Possible wider applications of opty

[Peter230655]

I have tried maybe 20 examples out of John T. Betts' book Practical Method for Optimal Control Using Nonlinear Programming, 3rd edition. Some are on pst-notebooks, some I pushed to examples-gallery, some I simply did not mange to get to converge.
The two major 'hindrances' to do more examples seem to be:

• opty cannot accept inequalities in the DAEs. In some cases this may be circumvented by adding state variables, see e.g. PR Betts 10 1 #269
• opty cannot accept (I believe) instance constraints or bounds which depend on the values of other state or control variables at some point in time (see also 'idea' # 264. In some cases this may be circumvented by adding state variables, see e.g. PR Examples 10.73 and 10.74 from Betts' book #271 Update as of 10-12-24: opty CAN accept instance constraints such as $x_2(t_0) = x_1(t_f)$. I have not checked this for bounds.

Update: It does not seem to work for bounds. If I set, say, bounds = {x1 = (u1(t0), u2(tf)} I get an error saying, as I understand it, that the values in (..., ...) must be floats.

[Peter230655]

Update:
I believe, inequalities of the form $g(x(t), u(t), params) \leq h_i \leq f(x(t), u(t), params), t_0 \leq t \leq t_f$ , x(t) state variables, u(t) controls, $h_i$ either a state or a control variable can be handled by introducing additional state variables $aux_i$, one for each inequality.
set $aux_i = h_i - g(...)$ as an additional equation in the eoms and force $aux_i \geq 0.0$ using the bounds, see e.g. PR #278, #277 for such simulations. ( of course $aux_j = f(...) - h_i$ for the other inequality)

if an inequality $f(x(t), u(t), params) \geq 0$ only holds for $t = t_f$ one can introduce an additional state variable $aux$, and additional unknown parameter $help$, and add $aux = g(...) - help$ to the eoms. Set $aux(t_f) = 0$ in the instance constraints, and bound $help > 0.0$. #271 is an example along these lines.

Of course, this crutch comes at the expense of enlarging the system.

[Peter230655]

To the best of my knowledge, opty does not allow inequalities in the instance constraints. A way around may be this:
Say x is a state variable, $t_m$ some time and one would want $a < x(t_m) < b$, with, for simplicity a, b > 0.
One way to do this:
introduce additional state variables top and bottom, and additional specifieds $top_h, bottom_h$.
add $top = top_h \cdot x$ and $bottom = bottom_h \cdot x$ to the equations of motion.
In the bounds set: { $top_h: (1, \infty), bottom_h: (0, 1)$ }, in the instance constraints set $top(t_m) - b, bottom(t_m) - a$.
An example is on pst-notebooks, called car_on_street_gate_opty.ipynb.

Again, this crutch enlarges the system substantially.

Update: if $t_0$ is the starting time of the simulation, the above does not work if $t_m = t_0$!
$t_m$ must be at least $t_0$ + interval_value, it seems.
What I am suggesting here is a crutch!

[Peter230655]

I pushed point_on_street_gate.ipynb to pst-notebooks.
It shows how to handle inequalities in the eoms and in the instance constraints.
I think, this may execute fast enough for examples-gallery.