Dear Carmine, only a few, simple optimal control problems are solvable in explicit form. The PMP system should be seen as a set of simultaneous conditions to be satisfied, but there exists no "algorithmic" way to solve it, although I would start by expressing the optimal control as a function of state and co-state via the minimum condition. One should also keep in mind that the optimal triple might not be unique.

What kind of optimal control problem do you have in mind?

Thank you for your answer prof. Ferretti.

I'm trying to realize an optimal control for an EV's regenerative braking, so I need to control a non-linear system (due to the model that describes the battery behavior) respecting three task with three different weights (following an acceleration profile, penalize the use of the hydraulic braking and maximize the generated power).

So, would it be more convenient change the approach and using antoher method?

Hi, Roberto Ferretti is completely true. The Hamiltonian is considered as a function of the control u, and time t, state x(t) and costate p(t) are parameters for the minimization step. In general, when the Hamiltonian is not too nonlinear, you can derive a closed-form solution u(t)=f(t,x(t),p(t)). Best

Dear Carmine, the problem seems way too complex to obtain a closed-form solution. I can suggest three approaches:

1. approach the PMP numerically;

2. direct minimization (i.e., treat the problem as a constrained minimization);

which have already been mentioned, and

3. dynamic programming, which provides an optimal feedback rather than an optimal open-loop solution. In this case, the state space has three dimensions, and this makes the problem feasible enough.