TITLE. An inverse problem in optimal control.

DESCRIPTION OF THE SUBJECT. (See first the proposed subject for “Laurea Triennale” Controllo ottimo). Let us suppose that we have a controlled ordinary differential system in Rn: y’=f(y,u), where u is the control at our disposal, and suppose that we have a control in feedback form u(t)=k(y(t)), that is we have a law, k, that tells us which is the control u to be used with only dependence on our actual position y(t) in Rn. Moreover, let us suppose that the control u should be chosen by a group of agents which, for instance, govern only a “piece” of the evolution y in Rn (for instance only one component), suppose that you are the system manger and that, for some particular reasons, your goal is to force every agents to use the feedback law k. One way to do that is, of course, to say to every agent: “use k!”, but often this is not a good strategy, since the agents are better to remain autonomous. A better strategy seems then to be the use of a sort of “incentive”, that is to find a goal for every agent (for instance to optimize a suitable performance) such that (as you, system manager, has previously checked) the right way to reach the goal is exactly use the law k. Hence, you leave the agents free to choose their control in order to reach the goal, but you know that, if they do the right thing, they use k.

This kind of situation is quite common (and studied) in several frameworks, such as for instance logistic optimization of distribution of a good through a network of stocks and stores. However, it seems that a rigorous study with the tools of mathematical analysis and optimal control theory is not yet done.

The approach for such a problem is to model the performance criterium (to be devolved to every agent) as the integral over the time of a cost function g depending on the state y(t) and on the control u(t) ,that is ò g(y(t),u(t))dt. The problem is then: given the controlled system y’=f(y,u) and the feedback control u*(t)=k(y(t)), find a function g such that the control u* is optimal for the cost ò g(y(t),u(t))dt, that is it minimizes the cost all over the possible controls. It is clear why this is an “inverse optimal control problem”: usually the control problem is: given the controlled system and the cost, find a control which is optimal. Here instead we already have a control and we want to find a cost in such a way the control turns out to be optimal.

GOAL OF THE THESIS. Using the Dynamic Programming Principle and the theory of Hamilton-Jacobi equations (which, if not yet known, may be assimilated as first part of the thesis work), try to write down a partial differential equation which, in some suitable sense, must be satisfied by the unknown cost g.

REQUIRED NOTIONS. The courses of mathematical analysis, the theory of ordinary differential equations, (suggested but not necessary) some notions on optimal control problems.