really final updates ICT

Mathematics/5th/Introduction_to_control_theory/Introduction_to_control_theory.tex

@@ -194,7 +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}. The last two equations are called the \emph{feedback control laws}.
+    The last two equations are called the \emph{feedback control laws}. 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
@@ -240,7 +240,7 @@
     where $L_aV(\vf{x}):=\pdv{V}{\vf{x}}\vf{a}(\vf{x})$ and $L_bV(\vf{x}):=\pdv{V}{\vf{x}}\vf{b}(\vf{x})$.
   \end{theorem}
   \begin{remark}
-    This $\psi$ is as smooth as $L_aV$ and $L_bV$ on $\RR^n\setminus\varnothing$. And if $V$ is a SCLF continuously at the origin, then $\vf\psi$ is continuous at the origin.
+    This $\psi$ is as smooth as $L_aV$ and $L_bV$ on $\RR^n\setminus\{0\}$. And if $V$ is a SCLF continuously at the origin, then $\vf\psi$ is continuous at the origin.
   \end{remark}
   \subsubsection{Backstepping}
   Consider a system of the form: