Whether Pontryagin's maximum principle valid or not?

Question

Let consider this system of the equation like $\dot{x}=p(1-e^{2x})$ where $x$ is position and $p$ is momentum. We want to minimize this payoff function $\int_{0}^{t}2a_1^2$ where $a_1\in[-\alpha,\alpha]$.
Now can we compute $H=f.w+r=p(1-e^{2x}).w+2a_1^2$ ? Note that $w$ is costate function and can we use Maximum principle to obtain time-optimal control?

Answer 1

If I understood correctly, $a_1$ is inconsequential on the dynamics of the system, therefore it will bind to zero along the trajectory in order to minimize the integral.

It seems also that the system is uncontrollable, which means that the trajectory is fully defined given the initial state.

The only decision variable left is the final time which is also inconsequential because $a_1$ will bind to zero. In case $a_1$ is forced to be always positive, for instance, both $a_1$ and $t_f$ will bind to their lower boundaries in order to minimize the integral.

Hope this helps.