TITLE. Differential games with exit-cost.

DESCRIPTION OF THE SUBJECT. A differential game is a problem of the following kind. Two players, let us say player X and player Y, may separately act on the trajectories of a dynamical system in Rⁿ in order to try to have their personal best response from the trajectory itself. The best response for player X is the minimization of a cost depending on the trajectory, whereas the best response for player Y is the maximization of the same cost: X and Y are then antagonists and a good performance for one of them is a bad performance for the other one.

Formally, a differential game may be written in the following way. Let us denote by a(·) and by b(·) the two strategies (depending on time) for player X and player Y, respectively (they are suitable functions taking value on suitable fixed sets of parameters A and B, respectively). The trajectory z(·) of the system, starting from an initial point z₀Î Rⁿ, is then given as the solution of the following system of ordinary differential equation in Rⁿ

z’(t)=f(z(t),a(t),b(t)) for t>0, z(0)= z₀.

Here we can see that the two strategies a and b directly act on the trajectory z (for different choices of a and b, the trajectory z is in general different). Moreover, the cost, which must depend on the initial state z₀ and on the strategies a and b, is usually given by the integral of a suitable running cost ℓ :

J(z₀,a,b)=òℓ(z(t),a(t),b(t),t)dt.

Player X wants to minimize J and player Y wants to maximize J. Of course, X does not know the currently strategy used by Y, and viceversa.

This kind of problems arises in many applications from economic, social, natural, and engineering sciences. The reader who has already given a look to the proposed subject for “Laurea Triennale” Controllo Ottimo will certainly understand that an optimal control problem is a particular kind of a differential game: it is a differential game with one player only, who wants to minimize the cost!

One of the major issue concerning differential games is to give a suitable notion of equilibrium, that is a criterium which is able to detect when two suitable strategies a’ and b’ make both players satisfied. In this setting, a suitable notion of equilibrium is the following one: let us define the functions

U(z₀)=max_b min_a J(z₀,a,b), V(z₀)=min_a max_b J(z₀,a,b).

Then if U=V, we can say that an equilibrium exists, and we can look for it. Using the so-called Dynamic Programming Method, the functions U and V are the unique solutions (in a suitable sense) of two first order non linear partial differential equations, respectively. Working on such equations, one can then give conditions in order to have U=V.

The Dynamic Programming Method is widely is used to study such a kind of problems. However, sometimes the cost is given in a slightly different way: it includes a so-called exit-cost. This means that the game lasts until the trajectory z stays inside a fixed domain WÌ Rⁿ. Let us denote by t’ the largest time such that z(t)ÎW for all tÎ[0,t’ ] (t’ is also called the exit-time), and let us suppose that we also have an exit-cost given by a function Y. In this case the cost J is given by

J(z₀,a,b)=ò₀ ^t’ ℓ(z(t),a(t),b(t),t)dt+Y(z(t’)),

that is: the cost is given by the integral up to the exit-time t’ plus an exit-cost Y paid on the exit-point z(t’ ). In this situation, which may arise in several applications (for instance when the trajectories are discontinuous, a so-called hybrid system) a major problem is the following: “which player is responsible of maintaining the trajectory inside W ?”, that is: “which player gains when the trajectory exits from W ?”. This kind of question is not well understood and treated in the literature, especially concerning the use of the Dynamic Programming Method and the study of the corresponding partial differential equations, where the function Y should enter as a boundary condition (in the same way for both equations?).

GOAL OF THE THESIS. The goal of the thesis is to formalize the exit-cost differential game problem and, in some particularly favourable cases, try to write down the corresponding partial differential equations satisfied by U and V, and study them.

REQUIRED NOTIONS. Cauchy problem for ordinary differential equations, measurable functions, differential and integral calculus for functions of several variables.