From 415449a6205f11bbe71bc1033d30937eca6a980a Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Mon, 22 Apr 2024 22:14:41 +0200 Subject: [PATCH] updated a lot intro control theory --- .../Introduction_to_control_theory.tex | 459 +++++++++++++++++- ...ntroduction_to_nonlinear_elliptic_PDEs.tex | 2 +- .../Stochastic_calculus.tex | 2 +- main_math.idx | 390 ++++++++------- main_math.ilg | 8 +- main_math.ind | 367 +++++++------- 6 files changed, 855 insertions(+), 373 deletions(-) diff --git a/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex b/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex index 6b49ed9..da1621d 100644 --- a/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex +++ b/Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex @@ -17,7 +17,7 @@ \begin{definition} Let $E\subseteq \RR^n$ be a neighbourhood of the origin and $V: E \to \RR_{\geq 0}$ be a function. We say that $V$ is \emph{positive definite} on $E$ if $\{V=0\} = \{0\}$. We say that $V$ is \emph{negative definite} on $E$ if $-V$ is positive definite on $E$. \end{definition} - \begin{lemma} + \begin{lemma}\label{ICT:lemmaK} Let $E\subseteq \RR^n$ be a neighbourhood of the origin and $V: E \to \RR_{\geq 0}$ be positive definite on $E$. Then, for any compact set $K \subseteq E$ with $0\in \Int K$, there exists $\alpha \in \mathcal{K}$ such that $\alpha(\norm{\vf{x}}) \leq V(\vf{x})$ for all $\vf{x} \in K$. \end{lemma} \begin{remark} @@ -25,7 +25,7 @@ $$ \abs{V(\vf{x}) - V(\vf{y})} \leq \omega(\norm{\vf{x}-\vf{y}}) $$ - where $\omega$ is a modulus of continuity of $V$. Then, we can find $\alpha_1 \in \mathcal{K}$ such that $\alpha_1\geq \omega$ and so we have an upper bound for $V(x)\leq \alpha_1(\norm{\vf{x}})$. + where $\omega$ is a modulus of continuity of $V$. Then, we can find $\alpha_1 \in \mathcal{K}^\infty$ such that $\alpha_1\geq \omega$ and so we have an upper bound for $V(x)\leq \alpha_1(\norm{\vf{x}})$. \end{remark} \begin{definition} Let $E\subseteq \RR^n$ be a neighbourhood of the origin. We defined the \emph{penalized norm} on $E$ as the function: @@ -64,7 +64,7 @@ Moreover, in the last two cases, if $\mu$ can be picked as large as we want, then the equilibrium is said to be \emph{globally stable}. \end{definition} \begin{remark} - Note that exponential stability implies asymptotic stability, which implies stability, which implies attractivity. Moreover, it can be seen that asymptotically stability is equivalent to stability and attractivity. + Note that exponential stability implies asymptotic stability, which implies stability and attractivity. Moreover, it can be seen that asymptotically stability is equivalent to stability and attractivity. \end{remark} \begin{remark} An equivalent definition for stability is the following: $\forall \varepsilon>0$ $\exists \delta>0$ such that if $\norm{\vf{x}_0}<\delta$ then $\norm{\vf{X}(\vf{x}_0, t)}<\varepsilon$ for all $t\geq 0$. @@ -92,7 +92,7 @@ We define the \emph{basin of attraction} of the origin as the set $\mathcal{A}$ of all initial conditions $\vf{x}_0$ such that the solution $\vf{X}(\vf{x}_0, \cdot)$ exists for all $t\geq 0$ and satisfies $\displaystyle\lim_{t\to\infty}\vf{X}(\vf{x}_0, t)=0$. \end{definition} \begin{theorem} - If the origin is asymptotically stable, then its basin of attraction is an open set included in $\mathcal{O}$. Besides, $\exists \beta_\mathcal{A}\in \mathcal{KL}$ such that $\forall \vf{x}_0\in\mathcal{A}$, any solution $\vf{X}(\vf{x}_0, \cdot)$ exists for all $t\geq 0$ and satisfies $\omega_{\mathcal{A}}(\norm{\vf{X}(\vf{x}_0, t)})\leq \beta_\mathcal{A}(\norm{\vf{x}_0}, t)$ for all $t\geq 0$. + If the origin is asymptotically stable, then its basin of attraction is an open set included in $\mathcal{O}$. Besides, $\exists \beta_\mathcal{A}\in \mathcal{KL}$ such that $\forall \vf{x}_0\in\mathcal{A}$, any solution $\vf{X}(\vf{x}_0, \cdot)$ exists for all $t\geq 0$ and satisfies $\omega_{\mathcal{A}}(\norm{\vf{X}(\vf{x}_0, t)})\leq \beta_\mathcal{A}(\norm{\vf{x}_0}, t)$ for all $t\geq 0$, where $\omega_{\mathcal{A}}$ is the penalized norm of $\mathcal{A}$. \end{theorem} \begin{theorem} Assume that $\vf{f}\in\mathcal{C}^1$. Then: @@ -101,6 +101,23 @@ \item If $\vf{Df}(\vf{0})$ has an eigenvalue with positive real part, then the origin is unstable. \end{enumerate} \end{theorem} + \begin{proof} + \begin{enumerate} + \item We only do the $\impliedby)$ part. So assume the origin is exponentially stable for the system $\dot{\vf{y}}=\vf{Df}(\vf{0}) \vf{y}$. Then, $\exists k,\lambda>0$ such that $\norm{\vf{y}(0,t)}\leq k\norm{\vf{y}_0}\exp{-\lambda t}$ for all $t\geq 0$, which implies $\exp{\vf{Df}(\vf{0})t}\leq k\exp{-\lambda t}$ for all $t\geq 0$. Now consider $\dot{\vf{x}}=\vf{f}(\vf{x})=\vf{Df}(\vf{0})\vf{x}+\Delta\vf{f}(\vf{x})$, with $\Delta\vf{f}(\vf{x}):=\vf{f}(\vf{x})-\vf{Df}(\vf{0})\vf{x}$. As $f\in\mathcal{C}^1$, $\exists R>0$ such that $\frac{\norm{\Delta\vf{f}(\vf{x})}}{\norm{\vf{x}}}\leq \frac{\lambda}{2 k}$ for all $\norm{\vf{x}}\leq R$. Defining $\mu:= \frac{R}{2k}$, then if $\norm{\vf{x}_0}\leq \mu$ we must have that the solution $\vf{X}(\vf{x}_0, \cdot)$ belongs to $B(\vf{0},R)$ at least on $[0,T)$ for certain $T>0$. Thus, $\forall t\in [0,T)$ we have $\frac{\norm{\Delta \vf{f}(\vf{X}(\vf{x}_0, t))}}{\norm{\vf{X}(\vf{x}_0, t)}}\leq \frac{\lambda}{2k}$, and so using the variations of constants formula: + $$ + \vf{X}(\vf{x}_0,t)=\exp{\vf{Df}(\vf{0})t}\vf{x}_0+\int_0^t\exp{\vf{Df}(\vf{0})(t-s)}\Delta\vf{f}(\vf{X}(\vf{x}_0,s))\dd s + $$ + Thus: + $$ + \norm{\vf{X}(\vf{x}_0,t)}\leq k \exp{-\lambda t}\norm{\vf{x}_0}+\frac{\lambda}{2}\int_0^t\exp{-\lambda(t-s)}\norm{\vf{X}(\vf{x}_0,s)}\dd s + $$ + And so: + $$ + \exp{\lambda t}\norm{\vf{X}(\vf{x}_0,t)}\leq k \norm{\vf{x}_0}+\frac{\lambda}{2}\int_0^t\exp{\lambda s}\norm{\vf{X}(\vf{x}_0,s)}\dd s + $$ + Finally, by \mnameref{SC:gronwall} we have $\exp{\lambda t}\norm{\vf{X}(\vf{x}_0,t)}\leq k \exp{\frac{\lambda}{2} t}\norm{\vf{x}_0}$, and so the origin is exponentially stable. + \end{enumerate} + \end{proof} \begin{remark} In linear dynamics exponentially stability is equivalent to global exponentially stability, which in turn is equivalent to global asymptotic stability which is equivalent to asymptotic stability. \end{remark} @@ -110,16 +127,16 @@ \begin{theorem} Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$. Then, if $$ - D_f^+V(\vf{x}) = \limsup_{t\to 0^+}\frac{V(\vf{x} + \vf{f}(\vf{x})t) - V(\vf{x})}{t} + D_f^+V(\vf{x}) = \limsup_{t\to 0^+}\frac{V(\vf{x} + t\vf{f}(\vf{x})) - V(\vf{x})}{t} $$ is non-positive for all $\vf{x}\in \mathcal{O}$, then the origin is stable. The function $V$ is called a \emph{Lyapunov function}. \end{theorem} \begin{proof} - Since $\mathcal{O}$ is a neighbourhood of the origin $\exists R>0$ such that $\overline{B(0,R)}\subseteq \mathcal{O}$. Then, since $V$ is continuous and positive definite, $\exists \alpha_1,\alpha_2\in\mathcal{K}$ such that $\alpha_1(\norm{\vf{x}})\leq V(\vf{x})\leq \alpha_2(\norm{\vf{x}})$ for all $\vf{x}\in B(0,R)$. Let $\mu:={\alpha_2}^{-1}(\alpha_1(R/2))$. Then, any solution with initial conditions $\norm{\vf{x}_0}<\mu$ belongs to $\overline{B(0,R)}$ at least for $t\in[0, T)$. Now if we consider $v(t) := V(\vf{X}(\vf{x}_0, t))$, then we have $\dot{v}(t) = D_f^+V(\vf{X}(\vf{x}_0, t))\leq 0$ for all $t\geq 0$. Thus, $\forall t \in [0, T)$ we have: + Since $\mathcal{O}$ is a neighbourhood of the origin $\exists R>0$ such that $\overline{B(0,R)}\subseteq \mathcal{O}$. Then, since $V$ is continuous and positive definite, $\exists \alpha_1,\alpha_2\in\mathcal{K}^\infty$ such that $\alpha_1(\norm{\vf{x}})\leq V(\vf{x})\leq \alpha_2(\norm{\vf{x}})$ for all $\vf{x}\in B(0,R)$ (by \cref{ICT:lemmaK}). Let $\mu:={\alpha_2}^{-1}(\alpha_1(R/2))$. Then, any solution with initial conditions $\norm{\vf{x}_0}<\mu$ belongs to $\overline{B(0,R)}$ at least for $t\in[0, T)$. Now if we consider $v(t) := V(\vf{X}(\vf{x}_0, t))$, then we have $\dot{v}(t) = D_f^+V(\vf{X}(\vf{x}_0, t))\leq 0$ for all $t\geq 0$. Thus, $\forall t \in [0, T)$ we have: \begin{multline*} \alpha_1(\norm{\vf{X}(\vf{x}_0, t)})\leq V(\vf{X}(\vf{x}_0, t))=v(t) \leq v(0)=\\ = V(\vf{x}_0) \leq \alpha_2(\norm{\vf{x}_0}) \end{multline*} - And so $\norm{\vf{X}(\vf{x}_0, t)}\leq \alpha_2^{-1}(\alpha_1(\norm{\vf{x}_0}))\leq R/2$ for all $t\in[0, T)$. This mean that in fact $T=\infty$ and so the origin is stable. + And so $\norm{\vf{X}(\vf{x}_0, t)}\leq \alpha_1^{-1}(\alpha_2(\norm{\vf{x}_0}))\leq R/2$ for all $t\in[0, T)$. This mean that in fact $T=\infty$ and so the origin is stable with the function $\alpha:=\alpha_1^{-1}\circ\alpha_2$. \end{proof} \begin{theorem} Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$. Then, if @@ -128,12 +145,25 @@ $$ with $w:\mathcal{O}\to \RR_{\geq 0}$ continuous and positive definite, then the origin is globally asymptotically stable. \end{theorem} + \begin{proof} + As in the previous proof, we define $\mu:=\alpha_2^{-1}(\alpha_1(R/2))$ and we get $\dot{v}(t)\leq -w(\vf{X}(\vf{x}_0, t))$. Since $w$ is continuous and positive definite, $\exists \alpha_3\in\mathcal{K}^\infty$ such that $\alpha_3(\norm{\vf{x}})\leq w(\vf{x})$ for all $\vf{x}\in \overline{B(0,R)}$. Thus, $\dot{v}(t) \leq -\alpha_3(\norm{\vf{X}(\vf{x}_0, t)})\leq -\alpha_3(\alpha_2^{-1}(V(\vf{X}(x_0, t))))$. Now, in this case, one can proove that $\exists \beta \in \mathcal{KL}$ such that $v(t)\leq \beta(v(0), t)$ for all $t\geq 0$. But: + \begin{multline*} + \alpha_1(\norm{\vf{X}(\vf{x}_0, t)})\leq V(\vf{X}(\vf{x}_0, t))=v(t) \leq \beta(v(0), t)=\\ = \beta(V(\vf{x}_0), t)\leq \beta(\alpha_2(\norm{\vf{x}_0}), t) + \end{multline*} + And thus, $\norm{\vf{X}(\vf{x}_0, t)}\leq \alpha_1^{-1}(\beta(\alpha_2(\norm{\vf{x}_0}), t))$ for all $t\geq 0$, which implies that the origin is globally asymptotically stable since the latter function is of class $\mathcal{KL}$. + \end{proof} \begin{theorem}[Lasaalle's invariance principle] Let $K$ be a compact set contained in $\mathcal{O}$ and let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function which is positive definite on $\mathcal{O}$ and such that $D_f^+V(\vf{x})\leq -w(\vf{x})$ for all $\vf{x}\in K$ with $w:\mathcal{O}\to \RR_{\geq 0}$ continuous (not necessarily positive definite). Then, for any solution $\vf{X}(\vf{x}_0, \cdot)$ with $\vf{x}_0\in K$ and defined on $K$ for all $t\geq 0$, $\exists v^*\in\RR_{\geq 0}$ such that $\vf{X}(\vf{x}_0, t)$ converges to the largest positively invariant set contained in: $$ \{\vf{y}\in K: V(\vf{y})=v^*\text{ and }w(\vf{y})=0\} $$ \end{theorem} + \begin{remark} + If the function $V$ is such that $$ + k_1\norm{\vf{x}}^n \leq V(\vf{x}) \leq k_2\norm{\vf{x}}^m + $$ + and $w$ such that $w(\norm{\vf{x}})\geq k_3\norm{\vf{x}}^m$, for some $k_1,k_2,k_3>0$ and $m,n\in\NN$, then the origin is globally exponentially stable. + \end{remark} \begin{theorem}[Chetaev's theorem] Let $V:\mathcal{O}\to \RR_{\geq 0}$ be a locally Lipschitz function such that: \begin{itemize} @@ -164,6 +194,7 @@ \dot{\vf{\vf\chi}} = \vf\psi(t, \vf{x}, \vf\chi) \end{cases} \end{equation} + If $q=0$, then the feedback control law is called \emph{static}, whereas if $q>0$ it is called \emph{dynamic}. Moreover if both $\vf\varphi$ and $\vf\psi$ are independent of $t$, then the control law is called \emph{stationary} and if $\vf\psi$ and $\vf\chi$ are independent of $\vf{x}$, it is called \emph{open-loop control}. \end{definition} \begin{theorem}[Kalmann's theorem] Consider the linear system $\dot{\vf{x}} = \vf{Ax} + \vf{Bu}$ with $\vf{A}\in \RR^{n\times n}$ and $\vf{B}\in \RR^{n\times p}$. Then, the system is controllable (or the pair $(\vf{A},\vf{B})$ is controllable) if and only if @@ -176,7 +207,7 @@ Let $\vf{A} \in \RR^{n\times n}$ and $\vf{B}\in \RR^{n\times p}$. Then, the pair $(\vf{A}, \vf{B})$ is controllable if and only if $\forall \lambda_1,\ldots,\lambda_n\in \CC$ $\exists \vf{K}\in \RR^{p\times n}$ such that: $$\sigma(\vf{A}+\vf{BK})=\{\lambda_1,\ldots,\lambda_n\}$$ \end{theorem} \begin{remark} - In practice we pick $\lambda_1,\ldots,\lambda_n\in \{\Re z<0\}$, and then we look for $\vf{K}$ such that $\sigma(\vf{A}+\vf{BK})\subseteq \{\lambda_1,\ldots,\lambda_n\}$ (for example by using the characteristic polynomial). Note that if $p>1$, the solution may not be unique. + In practice we pick $\lambda_1,\ldots,\lambda_n\in \{\Re z<0\}$, and then we look for $\vf{K}$ such that $\sigma(\vf{A}+\vf{BK})= \{\lambda_1,\ldots,\lambda_n\}$ (for example by using the characteristic polynomial). Note that if $p>1$, the solution may not be unique. \end{remark} \begin{theorem} Suppose that there exists $q\in \NN$, $\vf\psi:\RR^n\times \RR^q \to \RR^p$ and $\vf\varphi:\RR^n\times \RR^q \to \RR^q$ continuous such that $\vf\psi(\vf{0}, \vf{0}) = \vf{0}$ and $\vf\varphi(\vf{0}, \vf{0}) = \vf{0}$. Assume, moreover, that the system @@ -227,8 +258,418 @@ \end{cases} $$ \begin{lemma} - If $V$ is a $\mathcal{C}^2$ function and $\vf\eta$ is a $1/2$-Hölder continuous function, then $W(\vf{x}):=V(\vf{x}, \vf{\eta}(\vf{x}))$ is a SCLF for the system of \mcref{ICT:backstepping}. + If $V$ is a $\mathcal{C}^2$ function and $\vf\eta$ is a $1/2$-Hölder continuous function, then $W(\vf{x}):=V(\vf{x}, \vf{\eta}(\vf{x}))$ is a SCLF for the system $\dot{\vf{x}} = \vf{f}(\vf{x}, \vf{v})$. \end{lemma} + + Finally we consider $$ + V(\vf{x}, \vf{y}) = V(\vf{x}, \vf{\eta}(\vf{x})) +\int_{\vf{\eta}(\vf{x})}^{\vf{y}}\vf\varphi(\vf{x}, \vf{s})\dd s + $$ + with $\vf\varphi$ such that $\vf\varphi(\vf{x}, \vf{y}) = \vf{0}\iff \vf{y} = \vf{\eta}(\vf{x})$. Then, this $V$ is a SCLF for the system of \mcref{ICT:backstepping}. + \begin{remark} + Usually we take $\vf\varphi(\vf{x}, \vf{y}) = \vf{y}-\vf{\eta}(\vf{x})$ and consider + $$ + V(\vf{x}, \vf{y}) = W(\vf{x}) + \frac{1}{2}\left(\vf{y}-\vf{\eta}(\vf{x})\right)^2 + $$ + \end{remark} \subsection{Control theory in PDEs} + From what follows $\vf{x}$ will denote the state variable whose values are in a Hilbert space $\mathcal{X}$, and $\vf{u}$ will denote the control variable whose values are in a Hilbert space $\mathcal{U}$. + \subsubsection{Classical problems} + \begin{definition}[Exact controllability] + Let $T>0$. The \emph{exact controllability} of a system is said to be achieved if, for any initial condition $\vf{x}_0$ and any final condition $\vf{x}_T$, there exists a control $\vf{u}:[0,T]\to \mathcal{U}$ such that the solution $\vf{X}(\vf{x}_0, \cdot, \vf{u})$ of the system with initial condition $\vf{X}(\vf{x}_0, 0, \vf{u}) = \vf{x}_0$ satisfies $\vf{X}(\vf{x}_0, T, \vf{u}) = \vf{x}_T$. + \end{definition} + \begin{definition}[Approximate controllability] + Let $T>0$, $\varepsilon>0$. The \emph{approximate controllability} of a system is said to be achieved if, for any initial condition $\vf{x}_0$ and any final condition $\vf{x}_T$, there exists a control $\vf{u}:[0,T]\to \mathcal{U}$ such that the solution $\vf{X}(\vf{x}_0, \cdot, \vf{u})$ of the system satisfies $\norm{\vf{X}(\vf{x}_0, T, \vf{u}) - \vf{x}_T}<\varepsilon$. + \end{definition} + \begin{definition}[Null controllability] + Let $T>0$. The \emph{null controllability} of a system is said to be achieved if, for any initial condition $\vf{x}_0$, there exists a control $\vf{u}:[0,T]\to \mathcal{U}$ such that the solution $\vf{X}(\vf{x}_0, \cdot, \vf{u})$ of the system satisfies $\vf{X}(\vf{x}_0, T, \vf{u}) = \vf{0}$. + \end{definition} + \begin{lemma} + Consider a linear reversible system $\dot{\vf{x}} = \vf{A}\vf{x} + \vf{B}\vf{u}$ with $\vf{A}\in \RR^{n\times n}$ and $\vf{B}\in \RR^{n\times p}$. Then, the system is exactly controllable if and only if it is null controllable. + \end{lemma} + \begin{proof} + The implication to the right is clear. Now assume it is null controllable. Let $T>0$ and $x_0,x_T\in \RR^n$. Since the system is reversible we can first solve for $\overline{\vf{x}}$ + $$ + \begin{cases} + \dot{\overline{\vf{x}}} = \vf{A}\overline{\vf{x}} \\ + \overline{\vf{x}}(T) = \vf{x}_T + \end{cases} + $$ + Now we solve the null controllability problem with initial state $\vf{x}_0-\overline{\vf{x}}(0)$. Thus, we find $\vf{u}$ such that $\vf{x}$ satisfies + $$ + \begin{cases} + \dot{\vf{x}} = \vf{A}\vf{x} + \vf{B}\vf{u} \\ + \vf{x}(0) = \vf{x}_0-\overline{\vf{x}}(0) + \end{cases} + $$ + and so $\vf{x}(T) = 0$. Now consider $\widehat{\vf{x}}:= \overline{\vf{x}} + \vf{x}$. Then, $\widehat{\vf{x}}$ satisfies + $$ + \begin{cases} + \dot{\widehat{\vf{x}}} = \vf{A}\widehat{\vf{x}} + \vf{B}\vf{u} \\ + \widehat{\vf{x}}(0) = \vf{x}_0 \\ + \widehat{\vf{x}}(T) = \vf{x}_T + \end{cases} + $$ + \end{proof} + \begin{definition}[Feedback stabilization] + Given $\dot{\vf{x}}=\vf{Ax}+\vf{Bu}$, the \emph{feedback stabilization process} consists in finding an operator $K:\mathcal{X}\to\mathcal{U}$ such that $\dot{\vf{x}}=\vf{Ax}+\vf{B}\vf{K}\vf{x}$ has a stable (or asymptotically stable) equilibrium at the origin? + \end{definition} + \begin{definition}[Optimal control] + Let $J$ be a cost function, $J=J(\vf{x}, \vf{u},\vf{x}(T))$. The \emph{optimal control problem} consists in finding $\vf{u}: [0,T]\to \mathcal{U}$ such that $J$ is minimized, where $\vf{x}$ satisfies $\dot{\vf{x}} = \vf{Ax} + \vf{Bu}$ with $\vf{x}(0) = \vf{x}_0$. + \end{definition} + \subsubsection{Interior control for the heat equation} + Let $\Omega\subseteq \RR^n$ be a bounded regular domain (i.e.\ connected) and $\omega\subseteq \Omega$ be a non-empty open subset. We consider the control system: + \begin{equation}\label{ICT:heat_equation_control} + \begin{cases} + \partial_t v - \laplacian v = \indi{\omega}u & \text{in } [0,T] \times \Omega \\ + v = 0 & \text{in } [0,T] \times \Fr{\Omega} \\ + v = v_0 & \text{in } \Omega + \end{cases} + \end{equation} + \begin{theorem}[Strong solutions] + Let $f\in L^2((0,T);\Omega)$ and $v_0\in H_0^1(\Omega)$. Then, the Cauchy problem + \begin{equation}\label{ICT:heat_equation_cauchy} + \begin{cases} + \partial_t v - \laplacian v = f & \text{in } [0,T] \times \Omega \\ + v = 0 & \text{in } [0,T] \times \Fr{\Omega} \\ + v = v_0 & \text{in } \Omega + \end{cases} + \end{equation} + has a unique solution $$v\in \mathcal{C}^0([0,T]; H_0^1(\Omega))\cap L^2((0,T); H^2(\Omega)\cap H_0^1(\Omega))$$ + \end{theorem} + \begin{proof} + We start from uniqueness. Let $(e_i)_{i\in\NN}$ be a Hilbert basis of $L^2(\Omega)$ from the eigenvectors of the Laplacian operator: + $$ + \begin{cases} + -\laplacian e_i = \lambda_i e_i & \text{in } \Omega \\ + e_i = 0 & \text{in } \Fr{\Omega} + \end{cases} + $$ + and $\varphi\in \mathcal{D}((0,T)\times \Omega)$ be a test function. Then, we have + $$ + -\int_0^T\int_\Omega v \partial_t \varphi + \int_0^T\int_\Omega \nabla v \nabla \varphi = \int_0^T\int_\Omega f\varphi + $$ + In particular for $\varphi = \rho(t)\psi_{n,i}(x)$ with $\rho\in D(0,T)$ and $\psi_{n,i}\overset{H^1}{\to} e_i$ (here we use the fact that $H_0^1=\overline{\mathcal{D}(\Omega)}^{{}_{H^1}}$). Thus, we arrive at: + $$ + -\int_0^T\int_\Omega v \rho' e_i + \int_0^T\int_\Omega \rho\nabla v \nabla e_i = \int_0^T\int_\Omega f \rho e_i + $$ + Decomposing $v=\sum_{i\in\NN} v_i e_i$ and $f=\sum_{i\in\NN} f_i e_i$, we get: + $$ + -\int_0^T v_i \rho' + \lambda_i\int_0^T \rho v_i = \int_0^T f_i \rho + $$ + which in the sense of $D^*(0,T)$ gives $v_i'+\lambda_i v_i = f_i$, which has solution: + $$ + v_i(t)=\exp{-\lambda_i t}v_i(0)+\int_0^t\exp{-\lambda_i(t-s)}f_i(s)\dd s + $$ + So we have uniqueness and a formula: + \begin{align}\label{ICT:heat_equation_solution} + v(t,x) & =\sum_{i\in\NN}\exp{-\lambda_i t}v_i(0)e_i(x)+\sum_{i\in\NN}\int_0^t\exp{-\lambda_i(t-s)} f_i(s)e_i(x)\dd s \\ + & =:v_a(t,x)+v_b(t,x)\nonumber + \end{align} + For the existence, it suffices to check that the solution in \mcref{ICT:heat_equation_solution} belongs to the desired space. We first check that $v\in \mathcal{C}^0([0,T]; H_0^1(\Omega))$. We have: + \begin{multline*} + \norm{v_a}^2_{L^\infty(0,T; H_0^1(\Omega))} = \sup_{t\in[0,T]}\norm{v^a(t,\cdot)}_{H_0^1(\Omega)}^2 =\\=\sup_{t\in[0,T]}\int_\Omega\sum_{i\in\NN}\exp{-2\lambda_i t}\abs{v_i(0)}^2\norm{\grad e_i}^2=\\= \sup_{t\in[0,T]}\sum_{i\in\NN}\lambda_i\exp{-2\lambda_i t}\abs{v_i(0)}^2 \leq \sum_{i\in\NN}\lambda_i\abs{v_i(0)}^2 =\\= \norm{v_0}^2_{H_0^1(\Omega)} + \end{multline*} + On the other hand: + \begin{multline*} + \norm{v_b(t,\cdot)}^2_{H_0^1(\Omega)} = \sum_{i\in\NN}\lambda_i\left(\int_0^t\exp{-\lambda_i(t-s)}f_i(s)\dd s\right)^2=\\= \sum_{i\in\NN}\left( \int_0^t\exp{-\lambda_i(t-s)}f_i(s)\sqrt{\lambda_i}\dd s\right)^2 \leq \sum_{i\in\NN}\norm{f_i}_{L^2(0,T)}^2 = \\=\norm{f}^2_{L^2((0,T);L^2(\Omega))} + \end{multline*} + where we have used the fact that the penultimate term can be written as a convolution and then we use \mnameref{HA:youngConvolution} $\norm{f*g}_{L^r} \leq \norm{f}_{L^p}\norm{g}_{L^q}$ with $1/p+1/q=1+1/r$, in the case $p=q=2$ and $r=\infty$. Finally, we prove $v\in L^2((0,T); H^2(\Omega))$. Indeed: + \begin{multline*} + \norm{v_a(t,\cdot)}^2_{L^2(0,T;H^2)}=\int_0^T\sum_{i\in\NN}\lambda_i^2\exp{-2\lambda_i t}\abs{v_i(0)}^2\dd t=\\= \sum_{i\in\NN}\lambda_i\abs{v_i(0)}^2\int_0^T\lambda_i \exp{-2\lambda_i t}\dd t \leq C \norm{v_0}^2_{H_0^1(\Omega)} + \end{multline*} + because the latter term in the penultimate equality is bounded. Moreover: + \begin{multline*} + \norm{v_b(t,\cdot)}^2_{H_0^1} =\int_0^T\sum_{i\in\NN}\lambda_i^2 \left(\int_0^t\exp{-\lambda_i(t-s)}f_i(s)\dd s\right)^2\dd t=\\= \int_0^T \sum_{i\in\NN}\left(\int_0^t\exp{-\lambda_i(t-s)}f_i(s)\lambda_i\dd s\right)^2\dd t\leq\\\leq \int_0^T\sum_{i\in\NN}\norm{f_i}^2_{L^2(0,T)}\dd t \leq T \norm{f}^2_{L^2((0,T);L^2(\Omega))} + \end{multline*} + again by \mnameref{HA:youngConvolution}. + \end{proof} + \begin{theorem}[Weak solutions] + Let $f\in L^2((0,T);H^{-1}(\Omega))$ and $v_0\in L^2(\Omega)$. Then, the Cauchy problem of \mcref{ICT:heat_equation_cauchy} has a unique solution $$v\in \mathcal{C}^0((0,T); L^2(\Omega))\cap L^2((0,T); H_0^1(\Omega))$$ + \end{theorem} + We consider now the dual problem of \mcref{ICT:heat_equation_control}: + \begin{equation}\label{ICT:heat_equation_control_dual} + \begin{cases} + -\partial_t\theta - \laplacian\theta = 0 & \text{in } [0,T]\times \Omega \\ + \theta = 0 & \text{in } [0,T]\times \Fr{\Omega} \\ + \theta(T) = \theta_T & \text{in } \Omega + \end{cases} + \end{equation} + \begin{proposition}\label{ICT:duality} + Let $u\in L^2((0,T)\times \Omega)$, $v_0\in L^2(\Omega)$ and $v$ the corresponding solution of \mcref{ICT:heat_equation_control}. Then, the solution $\theta$ of \mcref{ICT:heat_equation_control_dual} with $\theta_T\in L^2(\Omega)$ satisfies: + $$ + \langle \theta, v\rangle_{L^2(\Omega)}\bigg|_0^T = \int_0^T\int_\Omega \indi{\omega}u \theta + $$ + \end{proposition} + \begin{proof} + We can sppose that all functions are smooth (otherwise we replace them by a linear combination of $e_i$ and pass to the limit using the fact that $(v_0,f)\mapsto v$ is continuous from $L^2(\Omega)\times L^2((0,T)\times \Omega)\to \mathcal{C}^0([0,T]; L^2(\Omega))\cap L^2((0,T); L^2(\Omega))$). Now, multiplying \mcref{ICT:heat_equation_control} by $\theta$ and integrating we get: + \begin{multline*} + \int_0^T\int_\Omega \indi{\omega}u\theta = \int_0^T\int_\Omega \partial_t v\theta + \int_0^T\int_\Omega \nabla v \nabla \theta=\\=\int_\Omega v\theta\bigg|_0^T-\int_0^T\int_\Omega \partial_t \theta v + \int_0^T\int_\Omega \nabla v \nabla \theta = \int_\Omega v\theta\bigg|_0^T + \end{multline*} + \end{proof} + \begin{definition} + We will say that the dual problem \mcref{ICT:heat_equation_control_dual} satisfies the \emph{finite-time observability inequality} if $\exists C>0$ such that $\forall \theta_T\in L^2(\Omega)$ the solution $\theta$ satisfies: + $$ + \norm{\theta(0)}_{L^2(\Omega)}^2\leq C\int_0^T\int_\omega \theta^2 + $$ + \end{definition} + \begin{proposition} + If the dual problem \mcref{ICT:heat_equation_control_dual} is finite-time observable, then the control problem \mcref{ICT:heat_equation_control} is null controllable. + \end{proposition} + \begin{proof} + Note that the null controllability condition is equivalent to $\forall \theta_T\in L^2(\Omega)$ we have (by \mnameref{ICT:duality}): + $$ + -\langle \theta(0),v_0\rangle_{L^2(\Omega)} = \int_0^T\int_\omega u \theta + $$ + Now let's define: + $$ + A:=\overline{\{\indi{\omega}\theta:\theta \text{ solution of \mcref{ICT:heat_equation_control_dual} for some }\theta_T\in L^2(\Omega)\}}^{{}_{L^2((0,T)\times \omega)}} + $$ + We equip $A$ with the norm $\norm{\cdot}_{L^2((0,T)\times \omega)}$. Now consider: + $$ + \function{\Phi}{L^2(\Omega)}{L^2((0,T)\times \omega)}{\theta_T}{\indi{\omega}\theta} + $$ + Note that $\overline{\Phi}=A$. Now, to any $\phi\in\im(\Phi)$ we could a priori associate several $\theta_T$, but all of them would generate the same $\theta_0$ due to the observability condition. So we may consider the map: + $$ + \function{}{\im(\Phi)}{L^2(\Omega)}{\indi{\omega}\theta}{\theta(0)} + $$ + which is continuous by the observability condition. Now, extending the map to $A$ by uniform continuity, we get that + $$ + \function{\ell}{A}{\RR}{\indi{\omega}\theta}{-\langle \theta(0),v_0\rangle_{L^2(\Omega)}} + $$ + is a continuous linear form (by composition). We conclude now with \mnameref{INEPDE:laxmilgram} since $A$ is a Hilbert space. + \end{proof} + \begin{proposition}[1D observability inequality] + Let $\Omega=(0,1)$, $\omega=(a,b)$ and $T>0$. Then, $\exists C>0$ such that $\forall \theta_T\in L^2(0,1)$ we have: + $$ + \norm{\theta(0)}_{L^2(0,1)}\leq C\norm{\theta}_{L^2((0,T)\times \omega)} + $$ + \end{proposition} + \begin{proof} + Let $w(t,x):=\theta(T-t,x)$ so that $w$ satisfies: + \begin{equation}\label{ICT:heat_equation_control_dual_1D} + \begin{cases} + \partial_t w - \partial_{xx} w = 0 & \text{in } [0,T]\times (0,1) \\ + w(t,0) = w(t,1) = 0 & \\ + w(0,x) = w_0(x) & \text{in } (0,1) + \end{cases} + \end{equation} + We want to prove that $$ + \norm{w(T)}_{L^2(0,1)}\leq C\norm{w}_{L^2((0,T)\times (a,b))} + $$ + From \mcref{ICT:lemma1Dobservability} between $t_1$ and $t_0$ we have: + $$ + \norm{w(t_0)}_{L^\infty(0,1)}\leq C\exp{\frac{D}{t_1-t_0}}\norm{w(t_1)}_{L^\infty(0,1)}^{1-\delta}\norm{w(t_0)}_{L^\infty(a,b)}^\delta + $$ + We repeat that in the interval $(t_2,t_1)$ and we get: + \begin{multline*} + \norm{w(t_0)}_{L^\infty(0,1)}\leq C^{2-\delta}\exp{\frac{D}{t_1-t_0}+\frac{D(1-\delta)}{t_2-t_1}}\norm{w(t_2)}_{L^\infty(0,1)}^{(1-\delta)^2}\cdot\\\cdot\norm{w(t_1)}_{L^\infty(0,1)}^{(1-\delta)\delta}\norm{w(t_0)}_{L^\infty(a,b)}^{\delta} + \end{multline*} + Repeating the argument we get each time an extra power $1-\delta$ in $\exp{D/(t_{n+1}-t_n)}$. So we would like to have for example $t_{n+1}-t_n=\alpha \left(1-\frac{\delta}{2}\right)^n$ $\sum_{n\in\NN} \left(1-\frac{\delta}{2}\right)^n = \frac{2}{\delta}$ so we let $t_0=T$ and $t_{n+1}=t_n-\frac{\delta}{2}T\left(1-\frac{\delta}{2}\right)^n$. We conclude arguing by induction and passing to the limit: + $$ + \norm{w(0)}_{L^\infty (0,1)}\leq C\norm{w}_{L^\infty((0,T)\times (a,b))} + $$ + Now to prove the $L^2$ inequality, for the left hand side we have $\norm{w(0)}_{L^2(0,1)}\leq \norm{w(0)}_{L^\infty(0,1)}$ and for the right hand side we use \mnameref{ICT:interiorregularity}. + \end{proof} + \begin{lemma}\label{ICT:lemma1Dobservability} + Using the hypotheses and notation of the previous proposition, we have that $\exists C,D>0$ and $\delta>0$ such that $\forall w_0$ we have: + $$ + \norm{w(T)}_{L^\infty(0,1)}\leq C\exp{D/T}\norm{w_0}_{L^\infty(0,1)}^{1-\delta}\norm{w(T)}_{L^\infty(a,b)}^\delta + $$ + \end{lemma} + \begin{lemma}[Interior regulariy]\label{ICT:interiorregularity} + Let $w$ be a solution of the heat equation ($w\in \mathcal{C}^0([0,T]; H_0^1(\Omega))\cap L^2((0,T); H^2(\Omega)\cap H_0^1(\Omega))$). Then: + $$ + \norm{w}_{L^\infty([T/2,T]\times [a,b])}\leq C\norm{w}_{L^2([T/4,T] \times [c,d])} + $$ + with $c0$: + $$ + \int_0^T\!\!\int_{\Fr{\Omega}}\! \abs{\partial_{\vf{n}}v}^2\! \leq\! C(1+T)\left[\norm{v_0}_{H_0^1}^2\!+\!\norm{v_1}_{L^2}^2\!+\!\norm{f}_{L^1(0,T;L^2(\Omega))}^2\right] + $$ + \end{theorem} + \begin{proof} + Suppose $v$ regular. We will use a multiplier method. Let $q:\overline{\Omega}\to \RR^n$ be a smooth vector field. We multiply the equation by $(\vf{q}\cdot \grad) v$ and integrate (we denote $Q:=[0,T]\times \Omega$ and $\Sigma_T:=[0,T]\times \Fr{\Omega}$). On the one hand: + \begin{multline*} + \iint_Q \partial_{tt} v (\vf{q}\cdot \grad) v = \int_\Omega\partial_tv (\vf{q}\cdot \grad) v\bigg|_0^T - \iint_Q \partial_t v (\vf{q}\cdot \grad) \partial_t v =\\= \int_\Omega\partial_tv (\vf{q}\cdot \grad) v\bigg|_0^T - \iint_Q (\vf{q}\cdot \grad) \frac{(\partial_t v)^2}{2} =\\= \int_\Omega\partial_tv (\vf{q}\cdot \grad) v\bigg|_0^T + \iint_Q \frac{(\partial_t v)^2}{2} \div \vf{q} + \end{multline*} + where in the last equality we have used intergation by parts and the \mnameref{FSV:divergencethm}. On the other hand: + \begin{multline*} + \iint_Q \laplacian v (\vf{q}\cdot \grad) v = \int_{\Sigma_T}\partial_{\vf{n}}v(\vf{q}\cdot \grad) v - \iint_Q \grad v \cdot \grad ((\vf{q}\cdot \grad) v) + \end{multline*} + Notice that since $v=0$ on $\Sigma_T$ the tangential derivatives of $v$ are zero, and so on $\Sigma_T$ we have $(\vf{q}\cdot \grad) v = (\vf{q}\cdot \vf{n})\partial_{\vf{n}}v$. + The second term can be written as: + \begin{multline*} + \iint_Q \grad v \cdot \grad ((\vf{q}\cdot \grad) v) = \iint_Q\partial_kv\partial_k (q_i\partial_iv)=\\=\iint_Q\partial_kv\partial_kq_i \partial_iv + \iint_Q\partial_kvq_i\partial_{ki}v=\\= \iint_Q\partial_kv\partial_kq_i \partial_iv + \iint_Qq_i\partial_i\left(\frac{(\partial_kv)^2}{2}\right)=\\= + \iint_Q\partial_kv\partial_kq_i \partial_iv + \iint_Q(\vf{q}\cdot \grad)\frac{\norm{\grad v}^2}{2}=\\= \iint_Q\partial_kv\partial_kq_i \partial_iv - \iint_Q\frac{\norm{\grad v}^2}{2}\div \vf{q}+\iint_{\Sigma_T}\frac{\norm{\grad v}^2}{2} \vf{q} \cdot \vf{n} + \end{multline*} + Finally grouping all terms we get: + \begin{multline*} + \frac{1}{2}\iint_{\Sigma_T} (\vf{q} \cdot \vf{n})(\partial_{\vf{n}}v)^2=\iint_Q\partial_kv\partial_kq_i \partial_iv -\frac{1}{2} \iint_Q \norm{\grad v}^2\div \vf{q} +\\+ \int_\Omega\partial_tv (\vf{q}\cdot \grad) v\bigg|_0^T + \iint_Q \frac{(\partial_t v)^2}{2} \div \vf{q} -\iint_Q f (\vf{q}\cdot \grad) v\lesssim\\ \lesssim a^2+a^2+ab+b+ac + \end{multline*} + with $a:=\norm{v}_{L^\infty([0,T]; H_0^1(\Omega))}$, $b:=\norm{\partial_t v}_{L^\infty([0,T]; L^2(\Omega))}$ and $c:=\norm{f}_{L^1((0,T); L^2(\Omega))}$. Here we used that $\norm{\grad v}=\abs{\partial_{\vf{n}}v}$, because $v=0$ on $\Sigma_T$. + We conclude choosing $\vf{q}$ a regular extension of the unit normal vector field to $\Omega$. + \end{proof} + Now, we want to define weak solutions of \mcref{ICT:wave_equation_control}. To do so, we take as test functions solutions of: + \begin{equation}\label{ICT:wave_equation_control_dual} + \begin{cases} + \partial_{tt} \theta - \laplacian \theta = f & \text{in } [0,T]\times \Omega \\ + \theta = 0 & \text{in } [0,T]\times \Fr{\Omega} \\ + (\theta,\partial_t \theta)|_{t=T} = (0,0) & \text{in } \Omega + \end{cases} + \end{equation} + \begin{definition}[Transposition solution] + Let $(v_0,v_1,u)\in L^2(\Omega)\times H^{-1}(\Omega)\times L^2(\Sigma_T)$. We call \emph{transposition solution} of \mcref{ICT:wave_equation_control} a function $v\in \mathcal{C}^0([0,T]; L^2(\Omega))\cap \mathcal{C}^1([0,T]; H^{-1}(\Omega))$ such that for any $f\in L^1((0,T); L^2(\Omega))$ we have: + \begin{equation}\label{ICT:transposition_solution} + \iint_Q vf = -\int_\Omega \partial_t\theta(0) v(0) +\int_\Omega \theta(0) v_1-\int_{\Sigma_T} u \partial_{\vf{n}} \theta + \end{equation} + where $\theta$ is the solution of \mcref{ICT:wave_equation_control_dual} associated to $f$. + \end{definition} + \begin{remark} + Any regular solution is a transposition solution. + \end{remark} + \begin{theorem} + For any $(v_0,v_1,u)\in L^2(\Omega)\times H^{-1}(\Omega)\times L^2(\Sigma_T)$, there exists a unique transposition solution of \mcref{ICT:wave_equation_control}. + \end{theorem} + \begin{proof} + We would like to prove that the right hand side of \mcref{ICT:transposition_solution} is a continuous linear form on $L^1((0,T); L^2(\Omega))$. If so then $\exists! v\in [L^1((0,T); L^2(\Omega))]^*=L^\infty([0,T]; L^2(\Omega))$ such the equation is true $\forall f\in L^1((0,T); L^2(\Omega))$. We have that + $$ + \function{}{L^1((0,T); L^2(\Omega))}{\mathcal{C}^0([0,T]; H_0^1(\Omega))\cap \mathcal{C}^1([0,T]; L^2(\Omega))}{f}{\theta} + $$ + is continuous, and so is $f\mapsto \int_\Omega \partial_t\theta(0) v(0)=\langle \partial_t\theta(0),v_0\rangle_{L^2\times L^2}$, because $\partial_t\theta(0)\in L^2(\Omega)$ and $v(0)\in L^2(\Omega)$. Similarly, since $f\to \theta(0)\in H_0^1(\Omega)$ is continuous, then so is $f\mapsto \int_\Omega \theta(0) v_1=\langle \theta(0),v_1\rangle_{H_0^1\times H^{-1}}$. Finally, by \mnameref{ICT:hiddenregularity} we have that $f\mapsto \partial_{\vf{n}}\theta|_{\Sigma_T}\in L^2$ is continuous, and so is $f\mapsto \int_{\Sigma_T} u \partial_{\vf{n}} \theta=\langle u,\partial_{\vf{n}}\theta\rangle_{L^2\times L^2}$. + \end{proof} + \begin{proposition} + Consider a transposition solution $v$ of \mcref{ICT:wave_equation_control}, with $v_0\in L^2(\Omega)$, $v_1\in H^{-1}(\Omega)$ and $u\in L^2((0,T)\times \Sigma)$. Let $\theta$ be a solution of + \begin{equation}\label{ICT:wave_equation_control_dual2} + \begin{cases} + \partial_{tt} \theta - \laplacian \theta =0 & \text{in } [0,T]\times \Omega \\ + \theta = 0 & \text{in } [0,T]\times \Fr{\Omega} \\ + (\theta,\partial_t \theta)|_{t=T} = (\theta_T^0,\theta_T^1) & \text{in } \Omega + \end{cases} + \end{equation} + Then: + $$ + \left[\langle \partial_tv,\theta\rangle_{H^{-1}\times H_0^1}-\langle v,\partial_t\theta\rangle_{L^2\times L^2}\right]\bigg|_0^T = \int_0^T\int_\Sigma u\partial_{\vf{n}}\theta + $$ + \end{proposition} + \begin{proof} + It is sufficient to prove it for regular solutions and then pass to the limit using: + \begin{align*} + \norm{\theta}_{L^\infty, H_0^1} + \norm{\partial_t\theta}_{L^\infty, L^2} & \lesssim \norm{\theta_T^0}_{H_0^1} + \norm{\theta_T^1}_{L^2} \\ + \norm{v}_{L^\infty, L^2}\!\!+\!\!\norm{\partial_t v}_{L^\infty, H^{-1}}\!\! & \lesssim \!\norm{v_0}_{L^2} \!\!+\!\! \norm{v_1}_{H^{-1}}\!\!+\!\!\norm{u}_{L^2((0,T)\times \Sigma)}\!\! + \end{align*} + Now, multiplying the equation of $v$ by $\theta$ and integrating we get: + \begin{multline*} + 0 = \int_0^T\int_\Omega (\partial_{tt}v -\laplacian v)\theta= \int_\Omega \partial_t v\theta\bigg|_0^T - \int_0^T\int_\Omega \partial_t v\partial_t\theta +\\+ \int_0^T\int_\Omega \nabla v \nabla \theta= \int_\Omega \partial_t v\theta\bigg|_0^T - \int_\Omega v \partial_t\theta\bigg|_0^T + \int_0^T\int_\Omega v \partial_{tt} \theta -\\- \int_0^T\int_\Omega v \laplacian \theta + \int_0^T\int_{\Fr{\Omega}} v\partial_{\vf{n}}\theta + \end{multline*} + \end{proof} + \begin{remark} + Exact controllability is equivalent to exact controllability starting from $(0,0)$ (due to superposition principle). + \end{remark} + \begin{definition}[Observability inequality] + We say that \mcref{ICT:wave_equation_control_dual2} is \emph{exactly observable} in time $T$ from $\Sigma$ if $\exists C>0$ such that for any solution $\theta$ of \mcref{ICT:wave_equation_control_dual2} we have: + $$ + \norm{\theta(0)}_{H_0^1}+\norm{\partial_t\theta(0)}_{L^2}\leq C\norm{\partial_{\vf{n}}\theta}_{L^2(\Sigma_T)} + $$ + \end{definition} + \begin{remark} + Note the difference with the final-time observability for the dual heat equation: + $$ + \norm{\theta(0)}_{L^2}\leq C\norm{\theta}_{L^2((0,T)\times \omega)} + $$ + \end{remark} + \begin{proposition} + If the dual problem \mcref{ICT:wave_equation_control_dual2} is exactly observable in time $T$ from $\Sigma$, then the control problem \mcref{ICT:wave_equation_control} is exactly controllable in time $T$ from $\Sigma$. + \end{proposition} + \begin{proof} + Suppose \mcref{ICT:wave_equation_control_dual2} is exactly observable. We make the choice to find $u$ of the form $u=\partial_{\vf{n}}\tilde{\theta}$ for some $\tilde{\theta}$ solution of \mcref{ICT:wave_equation_control_dual2} (in order to put the problem in the standard Riesz's form). Consider now $E:= H_0^1(\Omega)\times L^2(\Omega)$ equipped with the norm $\norm{(\theta_0,\theta_1)}_E:=\norm{\partial_{\vf{n}}\theta_0}_{L^2(\Sigma_T)}$, where $\theta$ is the solution of \mcref{ICT:wave_equation_control_dual2}. This is an equivalent norm to the standard one: + \begin{align*} + \norm{(\theta_0,\theta_1)}_E & \gtrsim \norm{\theta_0}_{H_0^1}+\norm{\theta_1}_{L^2} \text{(by observability inequality)} \\ + \norm{(\theta_0,\theta_1)}_E & \lesssim \norm{\theta_0}_{H_0^1}+\norm{\theta_1}_{L^2} \text{(by \mnameref{ICT:hiddenregularity})} + \end{align*} + $E$ is Hilber with this norm. Now, given $(\hat{v}_0,\hat{v}_1)\in E$, the left hand side is a continuous linear form on $(\theta(T),\partial_t\theta(T))\in E$. So $\exists (\overline{\theta}_0,\overline{\theta}_1)\in E$ such that with $\overline{\theta}$ the corresponding solution of \mcref{ICT:wave_equation_control_dual2} one has: $\forall (\theta(T),\partial_t\theta(T))\in E$ with the corresponding solution $\theta$ we have: + $$ + \langle \hat{v}_1,\theta(T)\rangle_{H^{-1}\times H_0^1}-\langle \hat{v}_0,\partial_t\theta(T)\rangle_{L^2\times L^2}=\int_{\Sigma_T}\partial_{\vf{n}}\overline{\theta}\partial_{\vf{n}}\theta + $$ + So we take $u:=\partial_{\vf{n}}\overline{\theta}$. + \end{proof} + \begin{theorem}[Bardos, Lebeau, Rauch] + The system is exactly controllable (or the dual observable) if and only if any ray of geometrical optics in $\Omega$ (at speed $1$) intersects $\Sigma$ between times $0$ and $T$. + \end{theorem} + \begin{remark} + If $\Sigma=\Fr{\Omega}$, then the system is controllable of $T>\diam(\Omega)$. + \end{remark} + \subsubsection{Abstract systems} \end{multicols} \end{document} \ No newline at end of file diff --git a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex index a7c82e0..ea206ef 100644 --- a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex +++ b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex @@ -282,7 +282,7 @@ Let $H$ be an infinite-dimensional separable Hilbert space and $K:H\to H$ be a compact operator. Then: \begin{enumerate} \item $0\in \sigma(K)$. - \item\label{INEPDE:item2_spectrum} If $\lambda\in \sigma(K)\setminus\{0\}$, then $\lambda$ is an eigenvalue of $K$. + \item\label{INEPDE:item2_spectrum} If $\lambda\in \sigma(K)\setminus\{0\}$, then $\lambda$ is an eigenvalue of $K$. \item $\sigma(K)$ is closed and at most countable. \item If $\sigma(K)\cap\RR$ is infinite, then $\sigma(K)\setminus\{0\}$ is of the form $\{\lambda_n\}_{n\in \NN}$ with $\lambda_n\to 0$. \item If $\lambda\in \sigma(K)\setminus\{0\}$, then: diff --git a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex index 8af0d96..3bb1f7c 100644 --- a/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex +++ b/Mathematics/5th/Stochastic_calculus/Stochastic_calculus.tex @@ -808,7 +808,7 @@ $$ \end{definition} \subsubsection{Existence and uniqueness of solutions} - \begin{lemma}[Gronwall's lemma]\label{SC:gronwall} + \begin{lemma}[Grönwall's lemma]\label{SC:gronwall} Let ${(x_t)}_{t\in[0,T]}$ be a non-negative function in $L^1([0,T])$ satisfying that $\forall t\in[0,T]$: $$ x_t\leq \alpha+\beta\int_0^t x_s\dd{s} diff --git a/main_math.idx b/main_math.idx index 644f8ef..6330f49 100644 --- a/main_math.idx +++ b/main_math.idx @@ -3178,189 +3178,211 @@ \indexentry{first order reflection|hyperpage}{352} \indexentry{Reillich-Kondrachov's compactness theorem|hyperpage}{352} \indexentry{trace operator|hyperpage}{352} -\indexentry{linear second-order PDE|hyperpage}{354} -\indexentry{non-divergence form|hyperpage}{354} -\indexentry{divergence form|hyperpage}{354} -\indexentry{Dirichlet boundary condition|hyperpage}{354} -\indexentry{homogeneous|hyperpage}{354} -\indexentry{Neumann boundary condition|hyperpage}{354} -\indexentry{homogeneous|hyperpage}{354} -\indexentry{uniformly elliptic|hyperpage}{354} -\indexentry{weak formulation|hyperpage}{354} -\indexentry{variational formulation|hyperpage}{354} -\indexentry{weak solution|hyperpage}{354} -\indexentry{strong solution|hyperpage}{354} -\indexentry{classical solution|hyperpage}{354} -\indexentry{continuous|hyperpage}{354} -\indexentry{coercive|hyperpage}{354} -\indexentry{symmetric|hyperpage}{354} -\indexentry{Lax-Milgram theorem|hyperpage}{355} -\indexentry{Abstract Fredholm alternative|hyperpage}{356} -\indexentry{formal adjoint|hyperpage}{356} -\indexentry{homogeneous adjoint problem|hyperpage}{356} -\indexentry{resolvent set|hyperpage}{357} -\indexentry{spectrum|hyperpage}{357} -\indexentry{precompact|hyperpage}{357} -\indexentry{Inner regularity|hyperpage}{357} -\indexentry{Regularity up to the boundary|hyperpage}{357} -\indexentry{Weak maximum principle|hyperpage}{358} -\indexentry{Weak maximum principle|hyperpage}{358} -\indexentry{Weak minimum principle|hyperpage}{358} -\indexentry{Schauder estimates|hyperpage}{359} -\indexentry{Weak maximum principle|hyperpage}{359} +\indexentry{class $\mathcal {K}$|hyperpage}{353} +\indexentry{class $\mathcal {K}^\infty $|hyperpage}{353} +\indexentry{class $\mathcal {KL}$|hyperpage}{353} +\indexentry{positive definite|hyperpage}{353} +\indexentry{negative definite|hyperpage}{353} +\indexentry{penalized norm|hyperpage}{353} +\indexentry{stable|hyperpage}{353} +\indexentry{attractive|hyperpage}{353} +\indexentry{asymptotically stable|hyperpage}{353} +\indexentry{exponentially stable|hyperpage}{353} +\indexentry{globally stable|hyperpage}{353} +\indexentry{basin of attraction|hyperpage}{353} +\indexentry{Lyapunov function|hyperpage}{354} +\indexentry{Lasaalle's invariance principle|hyperpage}{354} +\indexentry{Chetaev's theorem|hyperpage}{354} +\indexentry{controllable|hyperpage}{354} +\indexentry{asymptotically stabilizable|hyperpage}{354} +\indexentry{Kalmann's theorem|hyperpage}{354} +\indexentry{controllability matrix|hyperpage}{354} +\indexentry{strictly control Lyapunov function|hyperpage}{355} +\indexentry{SCLF|hyperpage}{355} +\indexentry{SCLF continuously at the origin|hyperpage}{355} +\indexentry{linear second-order PDE|hyperpage}{356} +\indexentry{non-divergence form|hyperpage}{356} +\indexentry{divergence form|hyperpage}{356} +\indexentry{Dirichlet boundary condition|hyperpage}{356} +\indexentry{homogeneous|hyperpage}{356} +\indexentry{Neumann boundary condition|hyperpage}{356} +\indexentry{homogeneous|hyperpage}{356} +\indexentry{uniformly elliptic|hyperpage}{356} +\indexentry{weak formulation|hyperpage}{356} +\indexentry{variational formulation|hyperpage}{356} +\indexentry{weak solution|hyperpage}{356} +\indexentry{strong solution|hyperpage}{356} +\indexentry{classical solution|hyperpage}{356} +\indexentry{continuous|hyperpage}{356} +\indexentry{coercive|hyperpage}{356} +\indexentry{symmetric|hyperpage}{356} +\indexentry{Lax-Milgram theorem|hyperpage}{357} +\indexentry{Abstract Fredholm alternative|hyperpage}{358} +\indexentry{formal adjoint|hyperpage}{358} +\indexentry{homogeneous adjoint problem|hyperpage}{358} +\indexentry{resolvent set|hyperpage}{359} +\indexentry{spectrum|hyperpage}{359} +\indexentry{precompact|hyperpage}{359} +\indexentry{Inner regularity|hyperpage}{359} +\indexentry{Regularity up to the boundary|hyperpage}{359} +\indexentry{Weak maximum principle|hyperpage}{360} +\indexentry{Weak maximum principle|hyperpage}{360} \indexentry{Weak minimum principle|hyperpage}{360} -\indexentry{Hopf's lemma|hyperpage}{360} -\indexentry{interior ball condition|hyperpage}{360} -\indexentry{Strong maximum principle|hyperpage}{360} -\indexentry{Strong minimum principle|hyperpage}{360} -\indexentry{A priori estimate|hyperpage}{360} -\indexentry{Continuation method|hyperpage}{360} -\indexentry{Brower fixed point|hyperpage}{361} -\indexentry{Schauder fixed point|hyperpage}{361} -\indexentry{convex hull|hyperpage}{361} -\indexentry{Schaefer fixed point|hyperpage}{361} -\indexentry{Without constraints|hyperpage}{362} -\indexentry{With constraints|hyperpage}{362} -\indexentry{Carathéodory|hyperpage}{363} -\indexentry{Superposition operator|hyperpage}{363} -\indexentry{superposition operator|hyperpage}{363} -\indexentry{Fréchet differentiable|hyperpage}{363} -\indexentry{Fréchet derivative|hyperpage}{363} -\indexentry{Gâteaux differentiable|hyperpage}{363} -\indexentry{Gâteaux derivative|hyperpage}{363} +\indexentry{Schauder estimates|hyperpage}{361} +\indexentry{Weak maximum principle|hyperpage}{361} +\indexentry{Weak minimum principle|hyperpage}{362} +\indexentry{Hopf's lemma|hyperpage}{362} +\indexentry{interior ball condition|hyperpage}{362} +\indexentry{Strong maximum principle|hyperpage}{362} +\indexentry{Strong minimum principle|hyperpage}{362} +\indexentry{A priori estimate|hyperpage}{362} +\indexentry{Continuation method|hyperpage}{362} +\indexentry{Brower fixed point|hyperpage}{363} +\indexentry{Schauder fixed point|hyperpage}{363} +\indexentry{convex hull|hyperpage}{363} +\indexentry{Schaefer fixed point|hyperpage}{363} \indexentry{Without constraints|hyperpage}{364} -\indexentry{coercive|hyperpage}{364} -\indexentry{Bootstrap|hyperpage}{364} -\indexentry{Lagrange multipliers|hyperpage}{364} -\indexentry{Lagrange multiplier|hyperpage}{365} -\indexentry{Lagrange multipliers in several variables|hyperpage}{365} -\indexentry{Lagrange multipliers|hyperpage}{365} -\indexentry{Aplication|hyperpage}{365} -\indexentry{Nehari manifold method|hyperpage}{365} -\indexentry{Nehari manifold method|hyperpage}{365} -\indexentry{Palais-Smale condition at level $c$|hyperpage}{366} -\indexentry{Ambrosetti-Rabinowitz theorem|hyperpage}{366} -\indexentry{Palais-Smale sequence|hyperpage}{366} -\indexentry{Mountain pass theorem|hyperpage}{366} -\indexentry{superquadradicity condition|hyperpage}{366} -\indexentry{Montecarlo estimator|hyperpage}{368} -\indexentry{seed|hyperpage}{368} -\indexentry{Mersenne Twister algorithm|hyperpage}{368} -\indexentry{Acceptance-rejection method|hyperpage}{368} -\indexentry{Box-Muller method|hyperpage}{369} -\indexentry{Polar method|hyperpage}{369} -\indexentry{antithetic method|hyperpage}{369} -\indexentry{control variate|hyperpage}{369} -\indexentry{multiple control variate estimator|hyperpage}{370} -\indexentry{equivalent|hyperpage}{370} -\indexentry{importance sampling estimator|hyperpage}{370} -\indexentry{Euler method|hyperpage}{371} -\indexentry{continuous Euler scheme|hyperpage}{371} -\indexentry{Strong error of the Euler scheme|hyperpage}{371} -\indexentry{Weak error of the Euler scheme|hyperpage}{371} -\indexentry{Romberg Extrapolation|hyperpage}{371} -\indexentry{Romberg Extrapolation|hyperpage}{371} -\indexentry{finite difference estimator|hyperpage}{372} -\indexentry{Black-Scholes model|hyperpage}{372} -\indexentry{tangent process|hyperpage}{372} -\indexentry{price of an American option|hyperpage}{373} -\indexentry{risk-free interest rate|hyperpage}{373} -\indexentry{discretization method|hyperpage}{373} -\indexentry{naive approach|hyperpage}{373} -\indexentry{Tsitsiklis-Van Roy method|hyperpage}{373} -\indexentry{Tsitsiklis-Van Roy method|hyperpage}{373} -\indexentry{Longstaff-Schwartz method|hyperpage}{374} -\indexentry{Longstaff-Schwartz method|hyperpage}{374} -\indexentry{Rogers's lemma|hyperpage}{374} -\indexentry{Dirichlet boundary conditions|hyperpage}{375} -\indexentry{Neumann boundary conditions|hyperpage}{375} -\indexentry{Robin boundary conditions|hyperpage}{375} -\indexentry{conforming Galerkin method|hyperpage}{375} -\indexentry{Céa's lemma|hyperpage}{375} -\indexentry{finite element|hyperpage}{375} -\indexentry{basis functions|hyperpage}{376} -\indexentry{local interpolant|hyperpage}{376} -\indexentry{subdivision|hyperpage}{376} -\indexentry{global interpolant|hyperpage}{376} -\indexentry{triangulation|hyperpage}{376} -\indexentry{affinely equivalent|hyperpage}{376} -\indexentry{Bramble-Hilbert lemma|hyperpage}{376} -\indexentry{diameter|hyperpage}{376} -\indexentry{insphere diameter|hyperpage}{376} -\indexentry{condition number|hyperpage}{376} -\indexentry{Local interpolation error|hyperpage}{377} -\indexentry{regular|hyperpage}{377} -\indexentry{Global interpolation error|hyperpage}{377} -\indexentry{spectral methods|hyperpage}{377} -\indexentry{trial functions|hyperpage}{377} -\indexentry{globally smooth|hyperpage}{377} -\indexentry{Galerkin methods|hyperpage}{377} -\indexentry{Collocation methods|hyperpage}{377} -\indexentry{collocation points|hyperpage}{377} -\indexentry{$\tau $ methods|hyperpage}{377} -\indexentry{interpolant|hyperpage}{377} -\indexentry{interpolant|hyperpage}{378} -\indexentry{jointly Gaussian|hyperpage}{379} -\indexentry{Brownian motion|hyperpage}{379} -\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{379} -\indexentry{Markov property for Brownian motion|hyperpage}{379} -\indexentry{natural filtration|hyperpage}{379} -\indexentry{Martingale|hyperpage}{379} -\indexentry{martingale|hyperpage}{379} -\indexentry{adapted|hyperpage}{379} -\indexentry{sub-martingale|hyperpage}{379} -\indexentry{super-martingale|hyperpage}{379} -\indexentry{hitting time|hyperpage}{380} -\indexentry{Doob's optional sampling theorem|hyperpage}{380} -\indexentry{stopped process|hyperpage}{380} -\indexentry{Orthogonality of martingales|hyperpage}{380} -\indexentry{Doob's maximal inequality|hyperpage}{380} -\indexentry{absoulte variation|hyperpage}{380} -\indexentry{finite variation|hyperpage}{380} -\indexentry{Quadratic variation|hyperpage}{380} -\indexentry{mesh|hyperpage}{380} -\indexentry{continuous local martingale|hyperpage}{381} -\indexentry{localizing sequence|hyperpage}{381} -\indexentry{Doob's optional sampling theorem for local martingales|hyperpage}{381} -\indexentry{Levy's characterization of Brownian motion|hyperpage}{382} -\indexentry{isometry|hyperpage}{382} -\indexentry{partial isometry|hyperpage}{382} -\indexentry{Wiener integral|hyperpage}{382} -\indexentry{Wiener isometry|hyperpage}{382} -\indexentry{Wiener integral|hyperpage}{382} -\indexentry{Wiener integral|hyperpage}{382} -\indexentry{Chasles relation|hyperpage}{382} -\indexentry{progressive|hyperpage}{383} -\indexentry{Itô integral|hyperpage}{384} -\indexentry{Itô isometry|hyperpage}{384} -\indexentry{Itô integral|hyperpage}{384} -\indexentry{generalized Itô integral|hyperpage}{384} -\indexentry{Stochastic dominated convergence theorem|hyperpage}{384} -\indexentry{Itô process|hyperpage}{385} -\indexentry{martingale term|hyperpage}{385} -\indexentry{drift term|hyperpage}{385} -\indexentry{stochastic differential|hyperpage}{385} -\indexentry{quadratic variation|hyperpage}{385} -\indexentry{Stochastic integration by parts|hyperpage}{386} -\indexentry{Itô term|hyperpage}{386} -\indexentry{Itô's formula|hyperpage}{386} -\indexentry{Doléans-Dade exponential|hyperpage}{386} -\indexentry{Novikov's condition|hyperpage}{386} -\indexentry{Giranov's theorem|hyperpage}{387} -\indexentry{drift|hyperpage}{387} -\indexentry{diffusion|hyperpage}{387} -\indexentry{stochastic differential equation|hyperpage}{387} -\indexentry{SDE|hyperpage}{387} -\indexentry{solution of the SDE|hyperpage}{387} -\indexentry{Gronwall's lemma|hyperpage}{387} -\indexentry{Existence and uniqueness of solutions of SDEs|hyperpage}{387} -\indexentry{Langevin equation|hyperpage}{387} -\indexentry{Geometric Brownian motion|hyperpage}{388} -\indexentry{Black-Scholes process|hyperpage}{388} -\indexentry{homogeneous SDE|hyperpage}{388} -\indexentry{diffusions|hyperpage}{388} -\indexentry{Invariance under time shift|hyperpage}{388} -\indexentry{Generator|hyperpage}{388} -\indexentry{generator|hyperpage}{388} -\indexentry{Kolmogorov's equation|hyperpage}{388} -\indexentry{Feynman-Kac's formula|hyperpage}{389} +\indexentry{With constraints|hyperpage}{364} +\indexentry{Carathéodory|hyperpage}{365} +\indexentry{Superposition operator|hyperpage}{365} +\indexentry{superposition operator|hyperpage}{365} +\indexentry{Fréchet differentiable|hyperpage}{365} +\indexentry{Fréchet derivative|hyperpage}{365} +\indexentry{Gâteaux differentiable|hyperpage}{365} +\indexentry{Gâteaux derivative|hyperpage}{365} +\indexentry{Without constraints|hyperpage}{366} +\indexentry{coercive|hyperpage}{366} +\indexentry{Bootstrap|hyperpage}{366} +\indexentry{Lagrange multipliers|hyperpage}{366} +\indexentry{Lagrange multiplier|hyperpage}{367} +\indexentry{Lagrange multipliers in several variables|hyperpage}{367} +\indexentry{Lagrange multipliers|hyperpage}{367} +\indexentry{Aplication|hyperpage}{367} +\indexentry{Nehari manifold method|hyperpage}{367} +\indexentry{Nehari manifold method|hyperpage}{367} +\indexentry{Palais-Smale condition at level $c$|hyperpage}{368} +\indexentry{Ambrosetti-Rabinowitz theorem|hyperpage}{368} +\indexentry{Palais-Smale sequence|hyperpage}{368} +\indexentry{Mountain pass theorem|hyperpage}{368} +\indexentry{superquadradicity condition|hyperpage}{368} +\indexentry{Montecarlo estimator|hyperpage}{370} +\indexentry{seed|hyperpage}{370} +\indexentry{Mersenne Twister algorithm|hyperpage}{370} +\indexentry{Acceptance-rejection method|hyperpage}{370} +\indexentry{Box-Muller method|hyperpage}{371} +\indexentry{Polar method|hyperpage}{371} +\indexentry{antithetic method|hyperpage}{371} +\indexentry{control variate|hyperpage}{371} +\indexentry{multiple control variate estimator|hyperpage}{372} +\indexentry{equivalent|hyperpage}{372} +\indexentry{importance sampling estimator|hyperpage}{372} +\indexentry{Euler method|hyperpage}{373} +\indexentry{continuous Euler scheme|hyperpage}{373} +\indexentry{Strong error of the Euler scheme|hyperpage}{373} +\indexentry{Weak error of the Euler scheme|hyperpage}{373} +\indexentry{Romberg Extrapolation|hyperpage}{373} +\indexentry{Romberg Extrapolation|hyperpage}{373} +\indexentry{finite difference estimator|hyperpage}{374} +\indexentry{Black-Scholes model|hyperpage}{374} +\indexentry{tangent process|hyperpage}{374} +\indexentry{price of an American option|hyperpage}{375} +\indexentry{risk-free interest rate|hyperpage}{375} +\indexentry{discretization method|hyperpage}{375} +\indexentry{naive approach|hyperpage}{375} +\indexentry{Tsitsiklis-Van Roy method|hyperpage}{375} +\indexentry{Tsitsiklis-Van Roy method|hyperpage}{375} +\indexentry{Longstaff-Schwartz method|hyperpage}{376} +\indexentry{Longstaff-Schwartz method|hyperpage}{376} +\indexentry{Rogers's lemma|hyperpage}{376} +\indexentry{Dirichlet boundary conditions|hyperpage}{377} +\indexentry{Neumann boundary conditions|hyperpage}{377} +\indexentry{Robin boundary conditions|hyperpage}{377} +\indexentry{conforming Galerkin method|hyperpage}{377} +\indexentry{Céa's lemma|hyperpage}{377} +\indexentry{finite element|hyperpage}{377} +\indexentry{basis functions|hyperpage}{378} +\indexentry{local interpolant|hyperpage}{378} +\indexentry{subdivision|hyperpage}{378} +\indexentry{global interpolant|hyperpage}{378} +\indexentry{triangulation|hyperpage}{378} +\indexentry{affinely equivalent|hyperpage}{378} +\indexentry{Bramble-Hilbert lemma|hyperpage}{378} +\indexentry{diameter|hyperpage}{378} +\indexentry{insphere diameter|hyperpage}{378} +\indexentry{condition number|hyperpage}{378} +\indexentry{Local interpolation error|hyperpage}{379} +\indexentry{regular|hyperpage}{379} +\indexentry{Global interpolation error|hyperpage}{379} +\indexentry{spectral methods|hyperpage}{379} +\indexentry{trial functions|hyperpage}{379} +\indexentry{globally smooth|hyperpage}{379} +\indexentry{Galerkin methods|hyperpage}{379} +\indexentry{Collocation methods|hyperpage}{379} +\indexentry{collocation points|hyperpage}{379} +\indexentry{$\tau $ methods|hyperpage}{379} +\indexentry{interpolant|hyperpage}{379} +\indexentry{interpolant|hyperpage}{380} +\indexentry{jointly Gaussian|hyperpage}{381} +\indexentry{Brownian motion|hyperpage}{381} +\indexentry{Strong law of large numbers for Brownian motion|hyperpage}{381} +\indexentry{Markov property for Brownian motion|hyperpage}{381} +\indexentry{natural filtration|hyperpage}{381} +\indexentry{Martingale|hyperpage}{381} +\indexentry{martingale|hyperpage}{381} +\indexentry{adapted|hyperpage}{381} +\indexentry{sub-martingale|hyperpage}{381} +\indexentry{super-martingale|hyperpage}{381} +\indexentry{hitting time|hyperpage}{382} +\indexentry{Doob's optional sampling theorem|hyperpage}{382} +\indexentry{stopped process|hyperpage}{382} +\indexentry{Orthogonality of martingales|hyperpage}{382} +\indexentry{Doob's maximal inequality|hyperpage}{382} +\indexentry{absoulte variation|hyperpage}{382} +\indexentry{finite variation|hyperpage}{382} +\indexentry{Quadratic variation|hyperpage}{382} +\indexentry{mesh|hyperpage}{382} +\indexentry{continuous local martingale|hyperpage}{383} +\indexentry{localizing sequence|hyperpage}{383} +\indexentry{Doob's optional sampling theorem for local martingales|hyperpage}{383} +\indexentry{Levy's characterization of Brownian motion|hyperpage}{384} +\indexentry{isometry|hyperpage}{384} +\indexentry{partial isometry|hyperpage}{384} +\indexentry{Wiener integral|hyperpage}{384} +\indexentry{Wiener isometry|hyperpage}{384} +\indexentry{Wiener integral|hyperpage}{384} +\indexentry{Wiener integral|hyperpage}{384} +\indexentry{Chasles relation|hyperpage}{384} +\indexentry{progressive|hyperpage}{385} +\indexentry{Itô integral|hyperpage}{386} +\indexentry{Itô isometry|hyperpage}{386} +\indexentry{Itô integral|hyperpage}{386} +\indexentry{generalized Itô integral|hyperpage}{386} +\indexentry{Stochastic dominated convergence theorem|hyperpage}{386} +\indexentry{Itô process|hyperpage}{387} +\indexentry{martingale term|hyperpage}{387} +\indexentry{drift term|hyperpage}{387} +\indexentry{stochastic differential|hyperpage}{387} +\indexentry{quadratic variation|hyperpage}{387} +\indexentry{Stochastic integration by parts|hyperpage}{388} +\indexentry{Itô term|hyperpage}{388} +\indexentry{Itô's formula|hyperpage}{388} +\indexentry{Doléans-Dade exponential|hyperpage}{388} +\indexentry{Novikov's condition|hyperpage}{388} +\indexentry{Giranov's theorem|hyperpage}{389} +\indexentry{drift|hyperpage}{389} +\indexentry{diffusion|hyperpage}{389} +\indexentry{stochastic differential equation|hyperpage}{389} +\indexentry{SDE|hyperpage}{389} +\indexentry{solution of the SDE|hyperpage}{389} +\indexentry{Gronwall's lemma|hyperpage}{389} +\indexentry{Existence and uniqueness of solutions of SDEs|hyperpage}{389} +\indexentry{Langevin equation|hyperpage}{389} +\indexentry{Geometric Brownian motion|hyperpage}{390} +\indexentry{Black-Scholes process|hyperpage}{390} +\indexentry{homogeneous SDE|hyperpage}{390} +\indexentry{diffusions|hyperpage}{390} +\indexentry{Invariance under time shift|hyperpage}{390} +\indexentry{Generator|hyperpage}{390} +\indexentry{generator|hyperpage}{390} +\indexentry{Kolmogorov's equation|hyperpage}{390} +\indexentry{Feynman-Kac's formula|hyperpage}{391} diff --git a/main_math.ilg b/main_math.ilg index 83d0e68..9a04c9f 100644 --- a/main_math.ilg +++ b/main_math.ilg @@ -1,6 +1,6 @@ -This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). -Scanning input file main_math.idx.......done (3366 entries accepted, 0 rejected). -Sorting entries.................................done (43903 comparisons). -Generating output file main_math.ind.......done (2866 lines written, 0 warnings). +This is makeindex, version 2.17 [TeX Live 2024] (kpathsea + Thai support). +Scanning input file main_math.idx.......done (3388 entries accepted, 0 rejected). +Sorting entries.................................done (44062 comparisons). +Generating output file main_math.ind.......done (2885 lines written, 0 warnings). Output written in main_math.ind. Transcript written in main_math.ilg. diff --git a/main_math.ind b/main_math.ind index bf461d8..841387d 100644 --- a/main_math.ind +++ b/main_math.ind @@ -40,7 +40,7 @@ \item $\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$, \hyperpage{175} \item $\sigma $-finite, \hyperpage{346} - \item $\tau $ methods, \hyperpage{377} + \item $\tau $ methods, \hyperpage{379} \item $d$-dimensional standard Brownian motion, \hyperpage{332} \item $i$-th pivot, \hyperpage{14} \item $k$-linear map, \hyperpage{155} @@ -78,7 +78,7 @@ \indexspace - \item A priori estimate, \hyperpage{360} + \item A priori estimate, \hyperpage{362} \item A-stable, \hyperpage{261} \item a.e., \hyperpage{293} \item Abel's summation formula, \hyperpage{77}, \hyperpage{106} @@ -96,9 +96,9 @@ \item absolutely convergent, \hyperpage{77}, \hyperpage{106}, \hyperpage{117}, \hyperpage{300} \item absolutely stable, \hyperpage{261} - \item absoulte variation, \hyperpage{380} - \item Abstract Fredholm alternative, \hyperpage{356} - \item Acceptance-rejection method, \hyperpage{368} + \item absoulte variation, \hyperpage{382} + \item Abstract Fredholm alternative, \hyperpage{358} + \item Acceptance-rejection method, \hyperpage{370} \item acceptation region, \hyperpage{199} \item ACCP, \hyperpage{46} \item accumulation point, \hyperpage{26}, \hyperpage{54} @@ -106,7 +106,7 @@ \item Adams method, \hyperpage{262} \item Adams-Bashforth method, \hyperpage{262} \item Adams-Moulton method, \hyperpage{262} - \item adapted, \hyperpage{348}, \hyperpage{379} + \item adapted, \hyperpage{348}, \hyperpage{381} \item adherence, \hyperpage{54} \item adherent point, \hyperpage{54}, \hyperpage{207} \item adjacency matrix, \hyperpage{50} @@ -121,7 +121,7 @@ \item affine plane, \hyperpage{66} \item affine space, \hyperpage{69} \item affine subvariety, \hyperpage{70} - \item affinely equivalent, \hyperpage{376} + \item affinely equivalent, \hyperpage{378} \item affinely independents, \hyperpage{69} \item affinity, \hyperpage{70} \item AIC, \hyperpage{257} @@ -143,7 +143,7 @@ \item alternating group, \hyperpage{42} \item alternating series, \hyperpage{77} \item alternative hypothesis, \hyperpage{199} - \item Ambrosetti-Rabinowitz theorem, \hyperpage{366} + \item Ambrosetti-Rabinowitz theorem, \hyperpage{368} \item amplification factor, \hyperpage{270} \item amplification polynomial, \hyperpage{273} \item Ampère's law, \hyperpage{278} @@ -156,9 +156,9 @@ \item angle-preserving, \hyperpage{149} \item anisotropic, \hyperpage{74} \item annihilator, \hyperpage{19} - \item antithetic method, \hyperpage{369} + \item antithetic method, \hyperpage{371} \item aperiodic, \hyperpage{320} - \item Aplication, \hyperpage{365} + \item Aplication, \hyperpage{367} \item approximation of identity, \hyperpage{79}, \hyperpage{241} \item approximations of the identity, \hyperpage{235} \item arc length, \hyperpage{60} @@ -183,11 +183,13 @@ \item asymptotic efficient estimator, \hyperpage{196} \item asymptotic error constant, \hyperpage{91} \item asymptotic line, \hyperpage{150} - \item asymptotically stable, \hyperpage{131} + \item asymptotically stabilizable, \hyperpage{354} + \item asymptotically stable, \hyperpage{131}, \hyperpage{353} \item asymptotically unbiased estimator, \hyperpage{194} \item atlas, \hyperpage{216} \item attracting, \hyperpage{131} \item attracting parabolic sector, \hyperpage{133} + \item attractive, \hyperpage{353} \item attractor fixed point, \hyperpage{91} \item auto-adjoint, \hyperpage{24} \item autonomous, \hyperpage{122} @@ -219,8 +221,9 @@ \item basic bootstrap confidence interval, \hyperpage{202} \item basic feasible solutions, \hyperpage{51} \item basin, \hyperpage{131} + \item basin of attraction, \hyperpage{353} \item basis, \hyperpage{16}, \hyperpage{206} - \item basis functions, \hyperpage{376} + \item basis functions, \hyperpage{378} \item Bautin's theorem, \hyperpage{226} \item Bayes estimate, \hyperpage{203} \item Bayes' formula, \hyperpage{174} @@ -253,8 +256,8 @@ \item bipartite, \hyperpage{50} \item birth and death process, \hyperpage{329} \item Bisection method, \hyperpage{90} - \item Black-Scholes model, \hyperpage{372} - \item Black-Scholes process, \hyperpage{388} + \item Black-Scholes model, \hyperpage{374} + \item Black-Scholes process, \hyperpage{390} \item block matrix, \hyperpage{15} \item blow-down, \hyperpage{225} \item Blow-up in cartesian coordinates, \hyperpage{226} @@ -268,7 +271,7 @@ \item Bonferroni's method, \hyperpage{252} \item Bonnet's theorem, \hyperpage{152} \item Boolean ring, \hyperpage{42} - \item Bootstrap, \hyperpage{364} + \item Bootstrap, \hyperpage{366} \item bootstrap distribution, \hyperpage{202} \item Bootstrap-t confidence interval, \hyperpage{202} \item bootstrap-t confidence interval, \hyperpage{202} @@ -287,12 +290,12 @@ \item bounded variation, \hyperpage{343} \item box topology, \hyperpage{209} \item Box-Cox transformation, \hyperpage{257} - \item Box-Muller method, \hyperpage{369} - \item Bramble-Hilbert lemma, \hyperpage{376} + \item Box-Muller method, \hyperpage{371} + \item Bramble-Hilbert lemma, \hyperpage{378} \item bridge, \hyperpage{50} \item Brouwer's fixed-point theorem, \hyperpage{215} - \item Brower fixed point, \hyperpage{361} - \item Brownian motion, \hyperpage{330}, \hyperpage{379} + \item Brower fixed point, \hyperpage{363} + \item Brownian motion, \hyperpage{330}, \hyperpage{381} \item Broyden's method, \hyperpage{263} \item Broyden-Fletcher-Goldfarb-Shanno method, \hyperpage{264} \item BTCS, \hyperpage{268} @@ -311,7 +314,7 @@ \item Cantor set, \hyperpage{209}, \hyperpage{342} \item Cantor's theorem, \hyperpage{9} \item Cantor-Bernstein theorem, \hyperpage{9} - \item Carathéodory, \hyperpage{363} + \item Carathéodory, \hyperpage{365} \item Cardano-Vieta's formulas, \hyperpage{114} \item cardinal, \hyperpage{6} \item Cartesian equations, \hyperpage{70} @@ -378,10 +381,11 @@ \item Characteristic function, \hyperpage{191} \item characteristic function, \hyperpage{9}, \hyperpage{191} \item characteristic polynomial, \hyperpage{20}, \hyperpage{49} - \item Chasles relation, \hyperpage{382} + \item Chasles relation, \hyperpage{384} \item Chebyshev method, \hyperpage{90} \item Chebyshev polynomials, \hyperpage{96} \item Chebyshev's inequality, \hyperpage{186}, \hyperpage{297} + \item Chetaev's theorem, \hyperpage{354} \item chi-squared distribution with $n$ degrees of freedom, \hyperpage{197} \item Chinese remainder theorem, \hyperpage{11} @@ -393,7 +397,10 @@ \item class $\mathcal {C}^n$ at a point $a\in \ensuremath {\mathbb {R}}$, \hyperpage{31} \item class $\mathcal {C}^p$, \hyperpage{31} - \item classical solution, \hyperpage{354} + \item class $\mathcal {KL}$, \hyperpage{353} + \item class $\mathcal {K}$, \hyperpage{353} + \item class $\mathcal {K}^\infty $, \hyperpage{353} + \item classical solution, \hyperpage{356} \item Classification of affine quadrics, \hyperpage{75} \item Classification of compact connected surfaces, \hyperpage{219} \item Classification of connected 1-manifolds, \hyperpage{217} @@ -430,13 +437,13 @@ \item codimension, \hyperpage{223} \item coefficient of determination, \hyperpage{252} \item coefficients, \hyperpage{11} - \item coercive, \hyperpage{354}, \hyperpage{364} + \item coercive, \hyperpage{356}, \hyperpage{366} \item cofactor, \hyperpage{136} \item cofactor matrix, \hyperpage{15} \item Cofinite topology, \hyperpage{206} \item collineation, \hyperpage{66} - \item Collocation methods, \hyperpage{377} - \item collocation points, \hyperpage{377} + \item Collocation methods, \hyperpage{379} + \item collocation points, \hyperpage{379} \item column rank, \hyperpage{16} \item Combinations with repetition, \hyperpage{10} \item Combinations without repetition, \hyperpage{10} @@ -486,7 +493,7 @@ \item computational efficiency, \hyperpage{91} \item concave, \hyperpage{30} \item Condensation test, \hyperpage{76} - \item condition number, \hyperpage{99}, \hyperpage{376} + \item condition number, \hyperpage{99}, \hyperpage{378} \item conditional consistency, \hyperpage{269} \item conditional expectation, \hyperpage{186} \item conditional expectation of $X$ given $\mathcal {G}$, @@ -508,7 +515,7 @@ \item configuration, \hyperpage{67} \item conformal, \hyperpage{119}, \hyperpage{149} \item conformal representation, \hyperpage{119} - \item conforming Galerkin method, \hyperpage{375} + \item conforming Galerkin method, \hyperpage{377} \item Congruence axioms, \hyperpage{63} \item congruence relation, \hyperpage{63} \item conic, \hyperpage{73} @@ -539,20 +546,20 @@ \item Construction of a non-SAS geometry, \hyperpage{66} \item contact, \hyperpage{144} \item contact of order $\geq n$ at $a$, \hyperpage{31} - \item Continuation method, \hyperpage{360} + \item Continuation method, \hyperpage{362} \item Continuity axioms, \hyperpage{63} \item Continuity correction, \hyperpage{192} \item continuity correction, \hyperpage{192} \item Continuity from above, \hyperpage{173} \item Continuity from below, \hyperpage{173} \item continuous, \hyperpage{28}, \hyperpage{55}, \hyperpage{106}, - \hyperpage{130}, \hyperpage{354} + \hyperpage{130}, \hyperpage{356} \item continuous at $x_0$, \hyperpage{28} \item Continuous equation, \hyperpage{277} \item continuous equation, \hyperpage{277} - \item continuous Euler scheme, \hyperpage{371} + \item continuous Euler scheme, \hyperpage{373} \item Continuous function, \hyperpage{208} - \item continuous local martingale, \hyperpage{381} + \item continuous local martingale, \hyperpage{383} \item Continuous memorylessness property, \hyperpage{178} \item Continuous uniform distribution, \hyperpage{178} \item continuous uniform distribution, \hyperpage{178} @@ -560,7 +567,9 @@ \item contractible, \hyperpage{139} \item contraction, \hyperpage{55}, \hyperpage{90} \item contrast matrix, \hyperpage{201} - \item control variate, \hyperpage{369} + \item control variate, \hyperpage{371} + \item controllability matrix, \hyperpage{354} + \item controllable, \hyperpage{354} \item converge in mean, \hyperpage{298} \item converge in norm $L^p$, \hyperpage{87} \item convergent, \hyperpage{26}, \hyperpage{54}, \hyperpage{76}, @@ -580,7 +589,7 @@ \item converges weakly-*, \hyperpage{349} \item convex, \hyperpage{30}, \hyperpage{115} \item convex functional, \hyperpage{308} - \item convex hull, \hyperpage{361} + \item convex hull, \hyperpage{363} \item convolution, \hyperpage{79}, \hyperpage{240}, \hyperpage{244} \item Cook's distance, \hyperpage{255} \item coordinate chart, \hyperpage{147}, \hyperpage{216} @@ -628,7 +637,7 @@ \item cyclotomic, \hyperpage{170} \item càd, \hyperpage{326} \item càdlàg, \hyperpage{176} - \item Céa's lemma, \hyperpage{375} + \item Céa's lemma, \hyperpage{377} \indexspace @@ -685,7 +694,7 @@ \item diagonal, \hyperpage{20} \item diagonalizable, \hyperpage{20} \item Diagonalization theorem, \hyperpage{20} - \item diameter, \hyperpage{376} + \item diameter, \hyperpage{378} \item dicyclic group, \hyperpage{42} \item diffeomorphism, \hyperpage{57} \item Differentiability criterion, \hyperpage{56} @@ -701,11 +710,11 @@ \item differential operator over distributions, \hyperpage{245} \item differential system, \hyperpage{122} \item differentiation operator, \hyperpage{285} - \item diffusion, \hyperpage{387} + \item diffusion, \hyperpage{389} \item diffusion coefficient, \hyperpage{278} \item Diffusion equation, \hyperpage{278} \item diffusion flux, \hyperpage{278} - \item diffusions, \hyperpage{388} + \item diffusions, \hyperpage{390} \item diffusivity, \hyperpage{278} \item digital topology, \hyperpage{206} \item dihedral group, \hyperpage{42} @@ -721,8 +730,8 @@ \item directional derivative, \hyperpage{55} \item director subspace, \hyperpage{70} \item Dirichlet, \hyperpage{274} - \item Dirichlet boundary condition, \hyperpage{354} - \item Dirichlet boundary conditions, \hyperpage{375} + \item Dirichlet boundary condition, \hyperpage{356} + \item Dirichlet boundary conditions, \hyperpage{377} \item Dirichlet kernel, \hyperpage{84}, \hyperpage{233} \item Dirichlet problem, \hyperpage{140}, \hyperpage{288} \item Dirichlet problem in the disc, \hyperpage{288} @@ -744,7 +753,7 @@ \item Discrete topology, \hyperpage{206} \item Discrete uniform distribution, \hyperpage{177} \item discrete uniform distribution, \hyperpage{177} - \item discretization method, \hyperpage{373} + \item discretization method, \hyperpage{375} \item discriminant, \hyperpage{169} \item displacement current, \hyperpage{278} \item distance, \hyperpage{53}, \hyperpage{82}, \hyperpage{205}, @@ -759,7 +768,7 @@ \hyperpage{241} \item distributional derivative, \hyperpage{242}, \hyperpage{285} \item divergence, \hyperpage{61} - \item divergence form, \hyperpage{354} + \item divergence form, \hyperpage{356} \item Divergence theorem, \hyperpage{160} \item Divergence theorem on $\ensuremath {\mathbb {R}}^2$, \hyperpage{62} @@ -767,22 +776,22 @@ \hyperpage{62}, \hyperpage{160} \item divergent, \hyperpage{26}, \hyperpage{76}, \hyperpage{105, 106} \item divided difference, \hyperpage{93} - \item Doléans-Dade exponential, \hyperpage{386} + \item Doléans-Dade exponential, \hyperpage{388} \item domain, \hyperpage{115}, \hyperpage{148} \item dominant eigenvalue, \hyperpage{100} \item dominant eigenvector, \hyperpage{100} \item Dominated convergence theorem, \hyperpage{184}, \hyperpage{189}, \hyperpage{297}, \hyperpage{346} - \item Doob's maximal inequality, \hyperpage{380} - \item Doob's optional sampling theorem, \hyperpage{380} + \item Doob's maximal inequality, \hyperpage{382} + \item Doob's optional sampling theorem, \hyperpage{382} \item Doob's optional sampling theorem for local martingales, - \hyperpage{381} + \hyperpage{383} \item dot product, \hyperpage{53} \item double, \hyperpage{90} \item Double dual space, \hyperpage{19} \item double dual space, \hyperpage{19} - \item drift, \hyperpage{387} - \item drift term, \hyperpage{385} + \item drift, \hyperpage{389} + \item drift term, \hyperpage{387} \item Du Bois-Reymond's test, \hyperpage{107} \item Du-Fort-Frankel scheme, \hyperpage{274} \item dual basis, \hyperpage{19} @@ -838,7 +847,7 @@ \item equivalence relation, \hyperpage{8} \item equivalent, \hyperpage{13}, \hyperpage{17}, \hyperpage{23}, \hyperpage{73}, \hyperpage{130}, \hyperpage{132}, - \hyperpage{306}, \hyperpage{370} + \hyperpage{306}, \hyperpage{372} \item equivalent dynamical systems, \hyperpage{131} \item Ergotic theorem, \hyperpage{324, 325} \item error of type I, \hyperpage{199} @@ -860,7 +869,7 @@ \item Euclidean vector space, \hyperpage{24} \item Euclidian division, \hyperpage{11} \item Euler characteristic, \hyperpage{218} - \item Euler method, \hyperpage{258}, \hyperpage{371} + \item Euler method, \hyperpage{258}, \hyperpage{373} \item Euler theorem, \hyperpage{50} \item Euler's formula, \hyperpage{108}, \hyperpage{150} \item Euler's theorem, \hyperpage{11} @@ -878,7 +887,7 @@ \item exact, \hyperpage{156} \item Examples of Euclidean motions, \hyperpage{73} \item Excluded point topology, \hyperpage{206} - \item Existence and uniqueness of solutions of SDEs, \hyperpage{387} + \item Existence and uniqueness of solutions of SDEs, \hyperpage{389} \item Existence of a lift, \hyperpage{337} \item Existence of orthogonal polynomials, \hyperpage{96} \item Existence of the rotation number, \hyperpage{338} @@ -902,6 +911,7 @@ \item exponential function with base $a$, \hyperpage{29} \item exponential generating function, \hyperpage{48} \item exponential series, \hyperpage{48} + \item exponentially stable, \hyperpage{353} \item extended complex plane, \hyperpage{105} \item extended real numbers, \hyperpage{105} \item extension, \hyperpage{126}, \hyperpage{308}, \hyperpage{351} @@ -932,7 +942,7 @@ \item FEM, \hyperpage{275} \item Fermat's little theorem, \hyperpage{11} \item Fermat's principle, \hyperpage{278} - \item Feynman-Kac's formula, \hyperpage{389} + \item Feynman-Kac's formula, \hyperpage{391} \item FFT, \hyperpage{240} \item Fick's law, \hyperpage{278} \item Fick's law of diffusion, \hyperpage{278} @@ -951,17 +961,17 @@ \item finite, \hyperpage{25}, \hyperpage{49}, \hyperpage{164}, \hyperpage{213}, \hyperpage{294} \item finite $k$-th moment, \hyperpage{185} - \item finite difference estimator, \hyperpage{372} + \item finite difference estimator, \hyperpage{374} \item finite difference method, \hyperpage{265} \item finite difference scheme, \hyperpage{268} - \item finite element, \hyperpage{375} + \item finite element, \hyperpage{377} \item finite element method, \hyperpage{275} \item finite expectation, \hyperpage{182--184} \item Finite field, \hyperpage{166} \item finite field, \hyperpage{166} \item finite moment of order $k$, \hyperpage{185} \item Finite subadditivity, \hyperpage{173} - \item finite variation, \hyperpage{380} + \item finite variation, \hyperpage{382} \item Finite-dimensional distributions, \hyperpage{333} \item finite-dimensional distributions, \hyperpage{333} \item finite-rank operator, \hyperpage{307} @@ -989,7 +999,7 @@ \item Flow box theorem, \hyperpage{132} \item flow of the linear ODE, \hyperpage{126} \item flux, \hyperpage{62}, \hyperpage{159} - \item formal adjoint, \hyperpage{356} + \item formal adjoint, \hyperpage{358} \item Formal derivative, \hyperpage{167} \item formal derivative, \hyperpage{167} \item Forward and backward difference formula of order 1, @@ -1012,8 +1022,8 @@ \item Fresnel integrals, \hyperpage{113} \item Frobenius endomorphism, \hyperpage{166} \item Fréchet, \hyperpage{212} - \item Fréchet derivative, \hyperpage{363} - \item Fréchet differentiable, \hyperpage{363} + \item Fréchet derivative, \hyperpage{365} + \item Fréchet differentiable, \hyperpage{365} \item FTBS, \hyperpage{268} \item FTCS, \hyperpage{268} \item FTFS, \hyperpage{268} @@ -1044,7 +1054,7 @@ \item Gagliardo, Nirengerg and Sobolev's inequality, \hyperpage{351} \item Galerkin approximation, \hyperpage{275} - \item Galerkin methods, \hyperpage{377} + \item Galerkin methods, \hyperpage{379} \item Galois, \hyperpage{168} \item Galois extension, \hyperpage{168} \item Galois group, \hyperpage{166} @@ -1081,35 +1091,36 @@ \item generalized eigenvector, \hyperpage{128} \item generalized heat kernel, \hyperpage{286} \item Generalized Hölder's inequality, \hyperpage{237} - \item generalized Itô integral, \hyperpage{384} + \item generalized Itô integral, \hyperpage{386} \item generalized solution, \hyperpage{245} \item generating set, \hyperpage{16} - \item Generator, \hyperpage{388} - \item generator, \hyperpage{388} + \item Generator, \hyperpage{390} + \item generator, \hyperpage{390} \item generator tree, \hyperpage{50} \item generatrix, \hyperpage{147} \item genus $g$ orientable surface, \hyperpage{217} \item genus $h$ non-orientable surface, \hyperpage{217} \item geodesic, \hyperpage{153} \item geodesic curvature, \hyperpage{153} - \item Geometric Brownian motion, \hyperpage{388} + \item Geometric Brownian motion, \hyperpage{390} \item Geometric distribution, \hyperpage{177} \item geometric distribution, \hyperpage{177} \item geometric multiplicity, \hyperpage{20} \item Gershgorin circle theorem, \hyperpage{92} - \item Giranov's theorem, \hyperpage{387} + \item Giranov's theorem, \hyperpage{389} \item Glide orthogonal reflections, \hyperpage{73} \item glide reflection, \hyperpage{72} \item glide vector, \hyperpage{73} \item Global Gau\ss -Bonnet theorem, \hyperpage{160} - \item global interpolant, \hyperpage{376} - \item Global interpolation error, \hyperpage{377} + \item global interpolant, \hyperpage{378} + \item Global interpolation error, \hyperpage{379} \item global maximum, \hyperpage{57} \item global minimum, \hyperpage{57} \item global stable manifold, \hyperpage{222} \item global truncation error, \hyperpage{258} \item global unstable manifold, \hyperpage{222} - \item globally smooth, \hyperpage{377} + \item globally smooth, \hyperpage{379} + \item globally stable, \hyperpage{353} \item Goodness of fit, \hyperpage{200} \item Goursat's theorem, \hyperpage{112} \item Gra\ss mann formula, \hyperpage{16}, \hyperpage{67} @@ -1126,14 +1137,14 @@ \item Green's formula, \hyperpage{159} \item Green's theorem, \hyperpage{62} \item grid, \hyperpage{268} - \item Gronwall's lemma, \hyperpage{387} + \item Gronwall's lemma, \hyperpage{389} \item Group, \hyperpage{37} \item group, \hyperpage{37} \item Group morphism, \hyperpage{38} \item group morphism, \hyperpage{38} \item Grönwall's lemma, \hyperpage{129} - \item Gâteaux derivative, \hyperpage{363} - \item Gâteaux differentiable, \hyperpage{363} + \item Gâteaux derivative, \hyperpage{365} + \item Gâteaux differentiable, \hyperpage{365} \indexspace @@ -1177,7 +1188,7 @@ \item Hilbert-Schmidt operator with kernel $K$, \hyperpage{307} \item Hilbert-Schmidt spectral representation theorem, \hyperpage{313} - \item hitting time, \hyperpage{380} + \item hitting time, \hyperpage{382} \item holding times, \hyperpage{327} \item holes, \hyperpage{216} \item holomorphic, \hyperpage{109} @@ -1188,14 +1199,14 @@ \item homoclinic bifurcation, \hyperpage{226} \item homoclinic orbit, \hyperpage{133} \item homogeneous, \hyperpage{49}, \hyperpage{122}, \hyperpage{126}, - \hyperpage{354} - \item homogeneous adjoint problem, \hyperpage{356} + \hyperpage{356} + \item homogeneous adjoint problem, \hyperpage{358} \item homogeneous coordinates, \hyperpage{67} \item Homogeneous distribution, \hyperpage{245} \item homogeneous of degree $k$, \hyperpage{221} \item homogeneous of degree $r\in \ensuremath {\mathbb {R}}$, \hyperpage{245} - \item homogeneous SDE, \hyperpage{388} + \item homogeneous SDE, \hyperpage{390} \item Homogenization, \hyperpage{74} \item homogenization, \hyperpage{70} \item homography, \hyperpage{67} @@ -1205,7 +1216,7 @@ \item homotopic, \hyperpage{139} \item homotopy, \hyperpage{139} \item Hopf bifurcation theorem, \hyperpage{226} - \item Hopf's lemma, \hyperpage{360} + \item Hopf's lemma, \hyperpage{362} \item Hopf-bifurcation, \hyperpage{226} \item Hurwitz's theorem, \hyperpage{119} \item hyperbolic, \hyperpage{268}, \hyperpage{273} @@ -1247,7 +1258,7 @@ \item implicit form, \hyperpage{122} \item Implicit function theorem, \hyperpage{57} \item Implicit scheme in finite differences, \hyperpage{286} - \item importance sampling estimator, \hyperpage{370} + \item importance sampling estimator, \hyperpage{372} \item improper, \hyperpage{37}, \hyperpage{203} \item improper integral, \hyperpage{79} \item in perspective with respect to a line, \hyperpage{68} @@ -1295,8 +1306,8 @@ \item initial values, \hyperpage{48} \item injective, \hyperpage{7} \item inner product, \hyperpage{24}, \hyperpage{82}, \hyperpage{310} - \item Inner regularity, \hyperpage{357} - \item insphere diameter, \hyperpage{376} + \item Inner regularity, \hyperpage{359} + \item insphere diameter, \hyperpage{378} \item integrable, \hyperpage{32}, \hyperpage{182, 183} \item integrable function, \hyperpage{59} \item integrable function over $E$, \hyperpage{297} @@ -1324,14 +1335,14 @@ \item inter-arrival times, \hyperpage{327} \item Interior, \hyperpage{206} \item interior, \hyperpage{54}, \hyperpage{206} - \item interior ball condition, \hyperpage{360} + \item interior ball condition, \hyperpage{362} \item interior point, \hyperpage{54}, \hyperpage{157}, \hyperpage{207} \item interior product, \hyperpage{156} \item Intermediate value theorem, \hyperpage{28}, \hyperpage{55}, \hyperpage{94}, \hyperpage{215} \item internally studentized residuals, \hyperpage{254} - \item interpolant, \hyperpage{377, 378} + \item interpolant, \hyperpage{379, 380} \item Interpolation inequality, \hyperpage{302} \item interpolation problem, \hyperpage{93} \item intersection, \hyperpage{6} @@ -1340,7 +1351,7 @@ \item Intervals for $\mu $ and $\sigma ^2$, \hyperpage{198} \item invariance level, \hyperpage{71} \item Invariance of the MLE, \hyperpage{195} - \item Invariance under time shift, \hyperpage{388} + \item Invariance under time shift, \hyperpage{390} \item invariant, \hyperpage{131}, \hyperpage{339, 340} \item invariant algebraic curve, \hyperpage{135} \item invariant subspace, \hyperpage{21} @@ -1371,7 +1382,7 @@ \item isolated singularity, \hyperpage{116}, \hyperpage{160} \item isometric, \hyperpage{23} \item isometry, \hyperpage{23}, \hyperpage{74}, \hyperpage{148}, - \hyperpage{382} + \hyperpage{384} \item isomorphic, \hyperpage{17}, \hyperpage{38}, \hyperpage{50}, \hyperpage{306} \item isomorphism, \hyperpage{38} @@ -1381,11 +1392,11 @@ \item iteration matrix, \hyperpage{99} \item itinerary, \hyperpage{230} \item Itinerary lemma, \hyperpage{229} - \item Itô integral, \hyperpage{384} - \item Itô isometry, \hyperpage{384} - \item Itô process, \hyperpage{385} - \item Itô term, \hyperpage{386} - \item Itô's formula, \hyperpage{386} + \item Itô integral, \hyperpage{386} + \item Itô isometry, \hyperpage{386} + \item Itô process, \hyperpage{387} + \item Itô term, \hyperpage{388} + \item Itô's formula, \hyperpage{388} \item ivp, \hyperpage{122} \indexspace @@ -1401,7 +1412,7 @@ \item joint probability density function, \hyperpage{180} \item Joint probability mass function, \hyperpage{180} \item joint probability mass function, \hyperpage{180} - \item jointly Gaussian, \hyperpage{379} + \item jointly Gaussian, \hyperpage{381} \item Jordan arc, \hyperpage{61} \item Jordan block, \hyperpage{21} \item Jordan closed curve, \hyperpage{61} @@ -1413,6 +1424,7 @@ \indexspace + \item Kalmann's theorem, \hyperpage{354} \item Kelley's theorem, \hyperpage{213} \item kernel, \hyperpage{17}, \hyperpage{38} \item kernel function, \hyperpage{79} @@ -1423,7 +1435,7 @@ \item Kolmogorov system, \hyperpage{221} \item Kolmogorov's backward equation, \hyperpage{328} \item Kolmogorov's continuity theorem, \hyperpage{331} - \item Kolmogorov's equation, \hyperpage{388} + \item Kolmogorov's equation, \hyperpage{390} \item Kolmogorov's forward equation, \hyperpage{328} \item Kolmogorov's strong law of large numbers, \hyperpage{191} \item Kronecker delta, \hyperpage{19} @@ -1437,17 +1449,18 @@ \item L'H\^opital's rule, \hyperpage{30} \item lack of fit test, \hyperpage{254} \item Lagrange basis polynomials, \hyperpage{93} - \item Lagrange multiplier, \hyperpage{365} - \item Lagrange multipliers, \hyperpage{364, 365} - \item Lagrange multipliers in several variables, \hyperpage{365} + \item Lagrange multiplier, \hyperpage{367} + \item Lagrange multipliers, \hyperpage{366, 367} + \item Lagrange multipliers in several variables, \hyperpage{367} \item Lagrange multipliers theorem, \hyperpage{58} \item Lagrange's interpolation problem, \hyperpage{93} \item Lagrange's theorem, \hyperpage{39} \item Laguerre polynomials, \hyperpage{96} \item Lamé coefficients, \hyperpage{277} - \item Langevin equation, \hyperpage{387} + \item Langevin equation, \hyperpage{389} \item Laplace equation, \hyperpage{140}, \hyperpage{288} \item Laplacian, \hyperpage{61} + \item Lasaalle's invariance principle, \hyperpage{354} \item lattice of subgroups, \hyperpage{169} \item Laurent series, \hyperpage{117} \item Laurent series theorem, \hyperpage{117} @@ -1457,7 +1470,7 @@ \item Law of total probability, \hyperpage{174}, \hyperpage{315} \item Lax theorem, \hyperpage{261} \item Lax-Friedrichs scheme, \hyperpage{268} - \item Lax-Milgram theorem, \hyperpage{355} + \item Lax-Milgram theorem, \hyperpage{357} \item Lax-Richtmyer equivalence theorem, \hyperpage{273} \item Lax-Wendroff, \hyperpage{271} \item Lax-Wendroff scheme, \hyperpage{271} @@ -1487,7 +1500,7 @@ \item level set, \hyperpage{55} \item leverage, \hyperpage{253} \item levorotation, \hyperpage{145} - \item Levy's characterization of Brownian motion, \hyperpage{382} + \item Levy's characterization of Brownian motion, \hyperpage{384} \item lexicographic degree, \hyperpage{163} \item lexicographic order, \hyperpage{163} \item lift, \hyperpage{337} @@ -1523,7 +1536,7 @@ \item linear programming to minimize, \hyperpage{51} \item linear recurrence relation of order $k$, \hyperpage{48} \item linear saddles, \hyperpage{227} - \item linear second-order PDE, \hyperpage{354} + \item linear second-order PDE, \hyperpage{356} \item linear span, \hyperpage{16} \item Linearity, \hyperpage{76} \item linearization, \hyperpage{221} @@ -1552,8 +1565,8 @@ \item local equivalence, \hyperpage{132} \item local extremum, \hyperpage{30}, \hyperpage{57} \item Local Gau\ss -Bonnet theorem, \hyperpage{160} - \item local interpolant, \hyperpage{376} - \item Local interpolation error, \hyperpage{377} + \item local interpolant, \hyperpage{378} + \item Local interpolation error, \hyperpage{379} \item local isometry, \hyperpage{148} \item local maximum, \hyperpage{30}, \hyperpage{57} \item local minimum, \hyperpage{30}, \hyperpage{57} @@ -1562,7 +1575,7 @@ \item local transversal section, \hyperpage{132} \item local truncation error, \hyperpage{262} \item local truncation errors, \hyperpage{258} - \item localizing sequence, \hyperpage{381} + \item localizing sequence, \hyperpage{383} \item locally, \hyperpage{214} \item locally bounded, \hyperpage{121}, \hyperpage{304} \item locally compact, \hyperpage{214} @@ -1580,7 +1593,7 @@ \item Logarithmic test, \hyperpage{76} \item logistic map, \hyperpage{228} \item logit, \hyperpage{257} - \item Longstaff-Schwartz method, \hyperpage{374} + \item Longstaff-Schwartz method, \hyperpage{376} \item loop, \hyperpage{49}, \hyperpage{215} \item Lorenz system, \hyperpage{228} \item loss function, \hyperpage{203} @@ -1595,7 +1608,7 @@ \item LU descompostion, \hyperpage{102} \item Lyapunov central limit theorem, \hyperpage{192} \item Lyapunov exponent, \hyperpage{337} - \item Lyapunov function, \hyperpage{137} + \item Lyapunov function, \hyperpage{137}, \hyperpage{354} \item Lyapunov stable, \hyperpage{137} \item Lyapunov's method, \hyperpage{226} \item Lyapunov's theorem, \hyperpage{137}, \hyperpage{223} @@ -1616,11 +1629,11 @@ \item marginal probability mass functions, \hyperpage{180} \item Markov chain, \hyperpage{318} \item Markov property, \hyperpage{318} - \item Markov property for Brownian motion, \hyperpage{379} + \item Markov property for Brownian motion, \hyperpage{381} \item Markov's inequality, \hyperpage{185} - \item Martingale, \hyperpage{379} - \item martingale, \hyperpage{348}, \hyperpage{379} - \item martingale term, \hyperpage{385} + \item Martingale, \hyperpage{381} + \item martingale, \hyperpage{348}, \hyperpage{381} + \item martingale term, \hyperpage{387} \item material derivative operator, \hyperpage{277} \item Matrix, \hyperpage{13} \item matrix, \hyperpage{13}, \hyperpage{18} @@ -1661,9 +1674,9 @@ \item Melnikov's method, \hyperpage{227} \item memoryless, \hyperpage{177, 178} \item meromorphic, \hyperpage{118} - \item Mersenne Twister algorithm, \hyperpage{368} + \item Mersenne Twister algorithm, \hyperpage{370} \item Mesh, \hyperpage{275} - \item mesh, \hyperpage{275}, \hyperpage{380} + \item mesh, \hyperpage{275}, \hyperpage{382} \item mesh-points, \hyperpage{258} \item Method of characteristics, \hyperpage{279} \item Method of moments, \hyperpage{194} @@ -1705,13 +1718,13 @@ \item monotonic, \hyperpage{26}, \hyperpage{28} \item monotonically decreasing, \hyperpage{26} \item monotonically increasing, \hyperpage{26} - \item Montecarlo estimator, \hyperpage{368} + \item Montecarlo estimator, \hyperpage{370} \item Montel's theorem, \hyperpage{121} \item more efficient than, \hyperpage{194} \item Morera's theorem, \hyperpage{114} \item morphism of field extensions, \hyperpage{165} \item Morrey's embedding, \hyperpage{351} - \item Mountain pass theorem, \hyperpage{366} + \item Mountain pass theorem, \hyperpage{368} \item MSE, \hyperpage{194} \item multi-index notation, \hyperpage{57} \item multicollinearity, \hyperpage{255} @@ -1720,7 +1733,7 @@ \item Multinomial distrbution, \hyperpage{180} \item multinomial distribution, \hyperpage{180} \item multiple, \hyperpage{10} - \item multiple control variate estimator, \hyperpage{370} + \item multiple control variate estimator, \hyperpage{372} \item multiple edges, \hyperpage{49} \item Multiple shooting method, \hyperpage{264} \item multiple shooting method, \hyperpage{264} @@ -1744,16 +1757,17 @@ \indexspace - \item naive approach, \hyperpage{373} + \item naive approach, \hyperpage{375} \item Nart-Vila theorem, \hyperpage{171} \item Natural cubic spline, \hyperpage{94} - \item natural filtration, \hyperpage{379} + \item natural filtration, \hyperpage{381} \item Navier-Cauchy equation, \hyperpage{277} \item negation, \hyperpage{8} \item negative, \hyperpage{141} \item negative basis, \hyperpage{142} \item Negative binomial distribution, \hyperpage{178} \item negative binomial distribution, \hyperpage{178} + \item negative definite, \hyperpage{353} \item negative orbit, \hyperpage{340} \item negative part, \hyperpage{77} \item negative semi-orbit, \hyperpage{130} @@ -1764,11 +1778,11 @@ \item negatively rotated, \hyperpage{227} \item negatively stable, \hyperpage{131} \item negatively-oriented, \hyperpage{142} - \item Nehari manifold method, \hyperpage{365} + \item Nehari manifold method, \hyperpage{367} \item neighbourhood, \hyperpage{25}, \hyperpage{54}, \hyperpage{207} \item Neumann, \hyperpage{274} - \item Neumann boundary condition, \hyperpage{354} - \item Neumann boundary conditions, \hyperpage{375} + \item Neumann boundary condition, \hyperpage{356} + \item Neumann boundary conditions, \hyperpage{377} \item Neumann series, \hyperpage{307} \item Neville's algorithm, \hyperpage{93} \item Newton method, \hyperpage{262}, \hyperpage{264} @@ -1790,7 +1804,7 @@ \item non-degenerate, \hyperpage{74} \item non-degenerated, \hyperpage{227} \item non-Desarguesian planes, \hyperpage{68} - \item non-divergence form, \hyperpage{354} + \item non-divergence form, \hyperpage{356} \item Non-parametric bootstrap, \hyperpage{202} \item Non-Paschian geometry, \hyperpage{65} \item Non-SAS geometry, \hyperpage{66} @@ -1821,7 +1835,7 @@ \item normed vector space, \hyperpage{53}, \hyperpage{300} \item not integrable, \hyperpage{182} \item not orientation-preserving, \hyperpage{217} - \item Novikov's condition, \hyperpage{386} + \item Novikov's condition, \hyperpage{388} \item null hypothesis, \hyperpage{199} \item null recurrent, \hyperpage{324} \item null set, \hyperpage{174}, \hyperpage{293} @@ -1901,7 +1915,7 @@ \item orthogonal system, \hyperpage{312} \item orthogonal with respect to the weight $\omega (x)$, \hyperpage{96} - \item Orthogonality of martingales, \hyperpage{380} + \item Orthogonality of martingales, \hyperpage{382} \item orthonormal, \hyperpage{22}, \hyperpage{82} \item orthonormal system, \hyperpage{82}, \hyperpage{312} \item orthonormalization, \hyperpage{313} @@ -1915,8 +1929,8 @@ \indexspace - \item Palais-Smale condition at level $c$, \hyperpage{366} - \item Palais-Smale sequence, \hyperpage{366} + \item Palais-Smale condition at level $c$, \hyperpage{368} + \item Palais-Smale sequence, \hyperpage{368} \item Paley-Wiener-Zygmund theorem, \hyperpage{331} \item Pappus configuration, \hyperpage{68} \item PAQ reduction theorem, \hyperpage{14} @@ -1942,7 +1956,7 @@ \item partial differential equation, \hyperpage{139} \item Partial fraction decomposition theorem, \hyperpage{117} \item partial inverse Fourier transform, \hyperpage{233} - \item partial isometry, \hyperpage{382} + \item partial isometry, \hyperpage{384} \item partial order relation, \hyperpage{9} \item Partial pivoting, \hyperpage{102} \item partially ordered set, \hyperpage{9} @@ -1963,6 +1977,7 @@ \item Peano axioms, \hyperpage{6} \item Peano theorem, \hyperpage{125} \item Pearson correlation coefficient, \hyperpage{185} + \item penalized norm, \hyperpage{353} \item Percentile confidence interval, \hyperpage{202} \item percentile confidence interval, \hyperpage{202} \item perfect, \hyperpage{166} @@ -2020,7 +2035,7 @@ \item Poisson summation formula, \hyperpage{238}, \hyperpage{240} \item Polar form, \hyperpage{108} \item polar form, \hyperpage{108} - \item Polar method, \hyperpage{369} + \item Polar method, \hyperpage{371} \item Polarization identity, \hyperpage{310} \item Pole, \hyperpage{116} \item pole, \hyperpage{116} @@ -2039,6 +2054,7 @@ \item poset, \hyperpage{9} \item positive, \hyperpage{141}, \hyperpage{158} \item positive basis, \hyperpage{142} + \item positive definite, \hyperpage{353} \item positive orbit, \hyperpage{340} \item positive part, \hyperpage{77} \item positive recurrent, \hyperpage{324} @@ -2063,14 +2079,14 @@ \item power series, \hyperpage{78} \item power set, \hyperpage{6} \item pre-Hilbert space, \hyperpage{310} - \item precompact, \hyperpage{357} + \item precompact, \hyperpage{359} \item predicate, \hyperpage{6} \item prediction, \hyperpage{253} \item prediction band, \hyperpage{254} \item preditions, \hyperpage{250} \item preimage, \hyperpage{7} \item preserves orientation, \hyperpage{337} - \item price of an American option, \hyperpage{373} + \item price of an American option, \hyperpage{375} \item primal, \hyperpage{51} \item prime, \hyperpage{10}, \hyperpage{12}, \hyperpage{44} \item Prime number theorem, \hyperpage{10} @@ -2102,7 +2118,7 @@ \item product measure, \hyperpage{346} \item product topology, \hyperpage{209} \item products of group subsets, \hyperpage{40} - \item progressive, \hyperpage{383} + \item progressive, \hyperpage{385} \item projection, \hyperpage{71} \item Projection theorem, \hyperpage{311} \item Projections, \hyperpage{71} @@ -2144,8 +2160,8 @@ \item Quadratic loss function, \hyperpage{203} \item quadratic polynomial, \hyperpage{162} \item quadratic space, \hyperpage{74} - \item Quadratic variation, \hyperpage{380} - \item quadratic variation, \hyperpage{385} + \item Quadratic variation, \hyperpage{382} + \item quadratic variation, \hyperpage{387} \item quadrature formula, \hyperpage{95} \item quadrature formula with weight $\omega (x)$, \hyperpage{97} \item quadric, \hyperpage{73} @@ -2220,12 +2236,12 @@ \item Regula falsi method, \hyperpage{90} \item regular, \hyperpage{49}, \hyperpage{58}, \hyperpage{131}, \hyperpage{141}, \hyperpage{195, 196}, \hyperpage{212}, - \hyperpage{328}, \hyperpage{377} + \hyperpage{328}, \hyperpage{379} \item regular distributions, \hyperpage{241} \item regular domain, \hyperpage{148} \item regular region, \hyperpage{160} \item regular surface, \hyperpage{147} - \item Regularity up to the boundary, \hyperpage{357} + \item Regularity up to the boundary, \hyperpage{359} \item Reillich-Kondrachov's compactness theorem, \hyperpage{352} \item Related samples with unknown variances, \hyperpage{199} \item relative condition numbers, \hyperpage{90} @@ -2244,7 +2260,7 @@ \item residuals, \hyperpage{250} \item residue, \hyperpage{116} \item Residues theorem, \hyperpage{117} - \item resolvent set, \hyperpage{357} + \item resolvent set, \hyperpage{359} \item response, \hyperpage{253} \item restrictions, \hyperpage{51} \item Reuter criterion, \hyperpage{330} @@ -2272,13 +2288,13 @@ \item ring, \hyperpage{42} \item Ring morphism, \hyperpage{43} \item ring morphism, \hyperpage{43} - \item risk-free interest rate, \hyperpage{373} + \item risk-free interest rate, \hyperpage{375} \item RK methods, \hyperpage{260} \item Robin, \hyperpage{274} - \item Robin boundary conditions, \hyperpage{375} - \item Rogers's lemma, \hyperpage{374} + \item Robin boundary conditions, \hyperpage{377} + \item Rogers's lemma, \hyperpage{376} \item Rolle's theorem, \hyperpage{30} - \item Romberg Extrapolation, \hyperpage{371} + \item Romberg Extrapolation, \hyperpage{373} \item Romberg method, \hyperpage{96} \item root, \hyperpage{12} \item root condition, \hyperpage{262} @@ -2318,19 +2334,21 @@ \item SAS criterion, \hyperpage{63, 64} \item scalars, \hyperpage{15} \item scale, \hyperpage{179} - \item Schaefer fixed point, \hyperpage{361} - \item Schauder estimates, \hyperpage{359} - \item Schauder fixed point, \hyperpage{361} + \item Schaefer fixed point, \hyperpage{363} + \item Schauder estimates, \hyperpage{361} + \item Schauder fixed point, \hyperpage{363} \item Schrödinger equation, \hyperpage{279} \item Schwartz space, \hyperpage{243} \item Schwarz lemma, \hyperpage{120} \item Schwarz theorem, \hyperpage{243} \item Schwarz's theorem, \hyperpage{57} \item Schwarz-Pick lemma, \hyperpage{121} + \item SCLF, \hyperpage{355} + \item SCLF continuously at the origin, \hyperpage{355} \item score function, \hyperpage{195} \item Score test, \hyperpage{202} \item score test, \hyperpage{202} - \item SDE, \hyperpage{387} + \item SDE, \hyperpage{389} \item Secant method, \hyperpage{90} \item secant-like method, \hyperpage{263} \item Second Borel-Cantelli lemma, \hyperpage{189} @@ -2341,7 +2359,7 @@ \item Second Sylow theorem, \hyperpage{41} \item section, \hyperpage{299} \item sectorial decomposition, \hyperpage{133} - \item seed, \hyperpage{368} + \item seed, \hyperpage{370} \item segment, \hyperpage{72} \item Segmented regression, \hyperpage{253} \item self-adjoint, \hyperpage{312} @@ -2432,7 +2450,7 @@ \item solution, \hyperpage{90}, \hyperpage{139} \item solution of a system of equations, \hyperpage{13} \item solution of the ODE, \hyperpage{122} - \item solution of the SDE, \hyperpage{387} + \item solution of the SDE, \hyperpage{389} \item solvable, \hyperpage{41}, \hyperpage{170} \item solvable by radicals, \hyperpage{170} \item SOR, \hyperpage{100} @@ -2440,11 +2458,11 @@ \item space of rapidly decreasing functions, \hyperpage{243} \item special orthogonal group, \hyperpage{145} \item specific heat capacity, \hyperpage{278} - \item spectral methods, \hyperpage{377} + \item spectral methods, \hyperpage{379} \item spectral radius, \hyperpage{98} \item Spectral theorem, \hyperpage{24}, \hyperpage{313} \item spectral values, \hyperpage{309} - \item spectrum, \hyperpage{97}, \hyperpage{309}, \hyperpage{357} + \item spectrum, \hyperpage{97}, \hyperpage{309}, \hyperpage{359} \item sphere, \hyperpage{54} \item Spline, \hyperpage{94} \item spline, \hyperpage{94} @@ -2458,7 +2476,7 @@ \item stability region, \hyperpage{261}, \hyperpage{268} \item stabilizer, \hyperpage{40} \item stable, \hyperpage{136}, \hyperpage{261}, \hyperpage{269}, - \hyperpage{273}, \hyperpage{328} + \hyperpage{273}, \hyperpage{328}, \hyperpage{353} \item stable degenerated node, \hyperpage{133} \item stable focus, \hyperpage{133} \item stable manifold, \hyperpage{222} @@ -2491,10 +2509,10 @@ \item stiff equations, \hyperpage{261} \item stiffness matrix, \hyperpage{276} \item Stirling's formula, \hyperpage{81} - \item stochastic differential, \hyperpage{385} - \item stochastic differential equation, \hyperpage{387} - \item Stochastic dominated convergence theorem, \hyperpage{384} - \item Stochastic integration by parts, \hyperpage{386} + \item stochastic differential, \hyperpage{387} + \item stochastic differential equation, \hyperpage{389} + \item Stochastic dominated convergence theorem, \hyperpage{386} + \item Stochastic integration by parts, \hyperpage{388} \item Stochastic matrix, \hyperpage{318} \item stochastic matrix, \hyperpage{318} \item Stochastic process, \hyperpage{316} @@ -2504,24 +2522,25 @@ \item Stokes' theorem, \hyperpage{62} \item Stolz-Cesàro theorem, \hyperpage{27} \item Stone-Weierstra\ss \ theorem, \hyperpage{304} - \item stopped process, \hyperpage{348}, \hyperpage{380} + \item stopped process, \hyperpage{348}, \hyperpage{382} \item stopping time, \hyperpage{321}, \hyperpage{348} \item strict Lyapunov function, \hyperpage{137} \item strictly concave, \hyperpage{30} + \item strictly control Lyapunov function, \hyperpage{355} \item strictly convex, \hyperpage{30} \item strictly decreasing, \hyperpage{26}, \hyperpage{28} \item strictly diagonally dominant by columns, \hyperpage{99} \item strictly diagonally dominant by rows, \hyperpage{99} \item strictly increasing, \hyperpage{26}, \hyperpage{28} \item Strong duality theorem, \hyperpage{52} - \item Strong error of the Euler scheme, \hyperpage{371} + \item Strong error of the Euler scheme, \hyperpage{373} \item Strong law of large numbers, \hyperpage{191} \item Strong law of large numbers for Brownian motion, - \hyperpage{379} + \hyperpage{381} \item Strong Markov property, \hyperpage{321, 322} - \item Strong maximum principle, \hyperpage{360} - \item Strong minimum principle, \hyperpage{360} - \item strong solution, \hyperpage{354} + \item Strong maximum principle, \hyperpage{362} + \item Strong minimum principle, \hyperpage{362} + \item strong solution, \hyperpage{356} \item strongly consistent estimator, \hyperpage{194} \item strongly lower-semicontinuous, \hyperpage{349} \item Structal stability, \hyperpage{336} @@ -2531,13 +2550,13 @@ \item Sturm's sequence, \hyperpage{92} \item Sturm's theorem, \hyperpage{92} \item Sturm-Picone comparison theorem, \hyperpage{283} - \item sub-martingale, \hyperpage{379} + \item sub-martingale, \hyperpage{381} \item sub-multiplicativity, \hyperpage{97} \item subadditive, \hyperpage{338} \item subalgebra, \hyperpage{303} \item subbasis, \hyperpage{206} \item subcover, \hyperpage{213} - \item subdivision, \hyperpage{376} + \item subdivision, \hyperpage{378} \item subfield generated, \hyperpage{163} \item Subgroup, \hyperpage{37} \item subgroup, \hyperpage{37} @@ -2568,12 +2587,12 @@ \hyperpage{107} \item sum of the series in an uniform sense, \hyperpage{78}, \hyperpage{107} - \item super-martingale, \hyperpage{379} + \item super-martingale, \hyperpage{381} \item superadditive, \hyperpage{338} \item supermartingale, \hyperpage{348} - \item Superposition operator, \hyperpage{363} - \item superposition operator, \hyperpage{363} - \item superquadradicity condition, \hyperpage{366} + \item Superposition operator, \hyperpage{365} + \item superposition operator, \hyperpage{365} + \item superquadradicity condition, \hyperpage{368} \item support, \hyperpage{8}, \hyperpage{79}, \hyperpage{146}, \hyperpage{176}, \hyperpage{180}, \hyperpage{242} \item support of $\mu $, \hyperpage{341} @@ -2592,7 +2611,7 @@ \item Sylow $p$-subgroup, \hyperpage{41} \item Sylvester's criterion, \hyperpage{58} \item Sylvester's law of inertia, \hyperpage{23} - \item symmetric, \hyperpage{22}, \hyperpage{62}, \hyperpage{354} + \item symmetric, \hyperpage{22}, \hyperpage{62}, \hyperpage{356} \item Symmetric difference formula of order 1, \hyperpage{95} \item Symmetric difference formula of order 2, \hyperpage{95} \item symmetric group, \hyperpage{8}, \hyperpage{38} @@ -2613,7 +2632,7 @@ \item tangent line to the graph at the point $(x_0,f(x_0))$, \hyperpage{29} \item tangent plane, \hyperpage{147} - \item tangent process, \hyperpage{372} + \item tangent process, \hyperpage{374} \item tangent space, \hyperpage{147}, \hyperpage{156, 157} \item tangent vector, \hyperpage{141}, \hyperpage{147}, \hyperpage{156} @@ -2699,14 +2718,14 @@ \item Trapezoidal rule, \hyperpage{95} \item traversable, \hyperpage{50} \item tree, \hyperpage{50} - \item trial functions, \hyperpage{377} + \item trial functions, \hyperpage{379} \item triangle, \hyperpage{160} \item Triangular inequality, \hyperpage{25}, \hyperpage{72} \item triangular inequality, \hyperpage{53}, \hyperpage{205}, \hyperpage{300} \item triangular system, \hyperpage{97} \item triangularization, \hyperpage{218} - \item triangulation, \hyperpage{376} + \item triangulation, \hyperpage{378} \item triple, \hyperpage{90} \item trivial $\sigma $-algebra, \hyperpage{172} \item Trivial topology, \hyperpage{206} @@ -2714,7 +2733,7 @@ \item truncated, \hyperpage{266} \item truncated Hilbert transform, \hyperpage{247} \item truncated singular value decomposition, \hyperpage{266} - \item Tsitsiklis-Van Roy method, \hyperpage{373} + \item Tsitsiklis-Van Roy method, \hyperpage{375} \item TSVD, \hyperpage{266} \item two times differentiable, \hyperpage{31} \item two times differentiable at $a$, \hyperpage{31} @@ -2739,7 +2758,7 @@ \item uniformly bounded, \hyperpage{125}, \hyperpage{304} \item uniformly continuous, \hyperpage{33}, \hyperpage{55} \item uniformly convergent, \hyperpage{117} - \item uniformly elliptic, \hyperpage{354} + \item uniformly elliptic, \hyperpage{356} \item uniformly equicontinuous, \hyperpage{125}, \hyperpage{304} \item uniformly integrable, \hyperpage{348} \item uniformly most powerful, \hyperpage{200} @@ -2786,7 +2805,7 @@ \item variance inflation factor, \hyperpage{255} \item Variation of constants, \hyperpage{123} \item variational equations, \hyperpage{129} - \item variational formulation, \hyperpage{275}, \hyperpage{354} + \item variational formulation, \hyperpage{275}, \hyperpage{356} \item Variations without repetition, \hyperpage{10} \item vector field, \hyperpage{60}, \hyperpage{152}, \hyperpage{154}, \hyperpage{156} @@ -2816,12 +2835,12 @@ \item wave equation, \hyperpage{140} \item wave function, \hyperpage{279} \item Weak duality theorem, \hyperpage{51} - \item Weak error of the Euler scheme, \hyperpage{371} - \item weak formulation, \hyperpage{275}, \hyperpage{354} + \item Weak error of the Euler scheme, \hyperpage{373} + \item weak formulation, \hyperpage{275}, \hyperpage{356} \item Weak law of large numbers, \hyperpage{190} - \item Weak maximum principle, \hyperpage{358, 359} - \item Weak minimum principle, \hyperpage{358}, \hyperpage{360} - \item weak solution, \hyperpage{354} + \item Weak maximum principle, \hyperpage{360, 361} + \item Weak minimum principle, \hyperpage{360}, \hyperpage{362} + \item weak solution, \hyperpage{356} \item weak-* lower-semicontinuity, \hyperpage{349} \item weakly consistent estimator, \hyperpage{194} \item weakly lower-semicontinuity, \hyperpage{349} @@ -2837,15 +2856,15 @@ \item well-ordered set, \hyperpage{9} \item well-posed, \hyperpage{258} \item well-posed in the Hadamard sense, \hyperpage{258} - \item Wiener integral, \hyperpage{382} - \item Wiener isometry, \hyperpage{382} + \item Wiener integral, \hyperpage{384} + \item Wiener isometry, \hyperpage{384} \item Wiener process, \hyperpage{330} \item winding number, \hyperpage{113} \item Wintner lemma, \hyperpage{126} \item Wirtinger operators, \hyperpage{111} \item Wirtinger's inequality, \hyperpage{87, 88} - \item With constraints, \hyperpage{362} - \item Without constraints, \hyperpage{362}, \hyperpage{364} + \item With constraints, \hyperpage{364} + \item Without constraints, \hyperpage{364}, \hyperpage{366} \item Witt's theorem, \hyperpage{74} \indexspace