# Pontryagin's maximum principle with two dimensional time

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?

• Are you talking about a partial differential equation? – Kwin van der Veen Dec 9 '17 at 23:07
• Kwin van der Veen I am maximizing something like $\int_{0}^{1} f[t,u(t), \int_{0}^{1} g(u(s),s)ds]dt$, where the control density function $u(\cdot )$ appears inside an integral. Constraints are imposed on non-negativity and the mean. – user391830 Dec 9 '17 at 23:44

## 1 Answer

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.

• Thanks a lot! This is very helpful! – user391830 Dec 10 '17 at 3:41
• Kwin van der Veen Sorry to bother again, but is there anything similar that I can do, if I were to maximize $\int_{0}^{T}g_{1}[t,x(t),u(t),\int_{0}^{T}g_{2}(t,s,u(s))ds]dt$ instead? – user391830 Dec 10 '17 at 22:10
• @user391830 The only thing I can think of would turn the problem into infinite dimensional problem by instead letting $a$ be a function in time, however the dimension of the extended state $z$ would then also change from one to infinite. – Kwin van der Veen Dec 10 '17 at 22:43
• @Kwin-van-der-Veen: regarding your solution, I believe that you should add $z(0)=0$. This would imply that you restrict the set of admissible controls to those that solve the two point BVP: $\dot{z}=g_2, z(0)=0, z(T)=a$, no? Such problems are not easy to solve... – Dmitry Dec 11 '17 at 6:32
• @Dmitry You are correct, however this does allow you to formulate the problem in a more standard way, for which there are techniques that can solve them. But I also think that this kind of problem is inherently difficult to solve. – Kwin van der Veen Dec 11 '17 at 6:52