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Existence and characterization of the values of two player differential games with state constraints

16 April 2019 at 3:00 PM - 4:00 PM

Room 105, Campus on Viale Romania 32

Speaker: Piernicola Bettiol, Univ. di Brest, France

Abstract:  Differential Games concern the balance of “optimal” strategies applied by two opposing players, who have conflicting notions of “best” performance of the dynamical system which they are both trying to control. Typically we are given an information pattern for the two players prescribing that each of them choose his own control at each instant of time without knowing the future choices of the opponent. This can be made rigorous by introducing the notion of non-anticipative strategy, establishing that each player knows the past and present behaviour of the other player.

We consider a two player, zero sum differential game with a cost of Bolza type, subject to a state constraint.  It is shown that, under a suitable hypothesis concerning existence of inward pointing velocity vectors for the minimizing player at the boundary of the constraint set, the lower value of the game is Lipschitz continuous and is the unique viscosity solution (appropriately defined) of the lower Hamilton Jacobi equation. If the inward pointing hypothesis is satisfied by the maximizing player's velocity set, then the upper game is Lipschitz continuous and is the unique solution of the upper Hamilton Jacobi equation.

Under the classical Isaacs condition, the upper and lower Hamilton Jacobi equations coincide. In this case, and when the inward pointing hypothesis is satisfied w.r.t. both players, the value of the game exists and is the unique solution to this Hamilton Jacobi equation. The novelty of our work resides in the fact that we permit the two players' controls to be completely coupled within the dynamic constraint, state constraint functional and the cost, in contrast to earlier work, in which the players' controls are decoupled w.r.t. the dynamics and state constraint, and interaction between them only occurs through the cost function. Furthermore, the inward pointing hypotheses that we impose are of a verifiable nature and less restrictive than those earlier employed.