Pontryagin's maximum principle with two dimensional time

Question

I'm working on a maximization of some double integral over distributions. Is there any results about the maximum principle in two-dimensional time with fixed time length and boundary conditions?

Answer 1

If you have an optimal control problem of the form

$$ \begin{align} \max_u & \int_0^T g_1\!\left(t,x(t),u(t),\int_0^T g_2(s,u(s))\,ds\right)\,dt \\ \textrm{s.t. } & \dot{x} = f(t,x,u) \\ & x_i(T) = c_i\ \forall\,i\in\mathcal{C}\subseteq \{1,2,\dots,n\} \end{align} $$

with $x\in\mathbb{R}^n$. This the problem could be reformulated to something that could be solved partially using Pontryagin's maximum principle. For this one would have to extend the state space by one and add constraints to that state. This then allows you to write the original problem as a nested optimisation problem

$$ \max_a \left[\begin{align} \max_u & \int_0^T g_1(t,x(t),u(t),a)\,dt \\ \textrm{s.t. } & \begin{bmatrix} \dot{x} \\ \dot{z} \end{bmatrix} = \begin{bmatrix} f(t,x,u) \\ g_2(t,u) \end{bmatrix} \\ & x_i(T) = c_i\ \forall\,i\in\mathcal{C}\subseteq \{1,2,\dots,n\} \\ & z(T) - z(0) = a \end{align}\right] $$

The inner optimisation problem can be solved for any given value of $a$ using PMP. And the outer optimisation problem is just a static optimisation problem, which hopefully is not too complex once you have solved for the general solution of the PMP problem.