Abstract
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-Isaacs 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-Isaacs equation. Under the classical Isaacs condition, the upper and lower Hamilton-Jacobi-Isaacs equation coincide. In this case, even if the inward pointing hypothesis is satisfied w.r.t. both players, the value of the game might fail to exist; however imposing stronger constraint qualifications (involving the existence of inward pointing vectors associated with saddle points for the Hamiltonian) the game value does exist and is the unique solution to this Hamilton-Jacobi-Isaacs 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 and the cost functional, 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.
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Acknowledgements
This research of the first and second authors benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research” (PGMO 2015-2832H, PGMO 2016-1570H), and from the support to this program from EDF and EDF-THALES-ORANGE-CRITEO. The research of the second author benefited from the support of the Project PGMO 2018-0047H, and was also partially supported by Research Contract AFOSR-FA9550-18-1-0254.
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Bettiol, P., Quincampoix, M. & Vinter, R.B. Existence and Characterization of the Values of Two Player Differential Games with State Constraints. Appl Math Optim 80, 765–799 (2019). https://doi.org/10.1007/s00245-019-09608-8
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DOI: https://doi.org/10.1007/s00245-019-09608-8