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Set-valued numerical analysis for optimal control and differential games. (English) Zbl 0982.91014

Bardi, Martino (ed.) et al., Stochastic and differential games. Theory and numerical methods. Dedicated to Prof. A. I. Subbotin. Boston: Birkhäuser. Ann. Int. Soc. Dyn. Games. 4, 177-247 (1999).
The following two-person zero-sum differential games are considered: the games with one target and opposite goals where the first player aims at reaching the target while the second player aims at avoiding the target (qualitative) or the first player aims at reaching the target in a minimal time while the second player aims at avoiding the target as long as possible (quantitative). In the first games the problem is to find the victory domain for each player, i.e., the set of initial conditions, starting from which this player succeeds to reach his goal whatever the opponent plays. The second games are pursuit-evasion games resulted in minimal hitting time problem. The two problems are treated in the framework of set-valued analysis and viability theory. To solve the problems the authors use the approximation of the viability kernel and discriminating kernel. General sufficient conditions for the pointwise convergence of a numerical scheme are given. Also the iterative algorithms avoid redoing computation over the whole initial domain at change of discretization step.
A similar approach is used for qualitative and quantitative control problems with state constraints. This gives the base of analysis. To approximate the viability kernel the authors consider the following steps: a time discretization through a Euler scheme, discretization both in time and in space (where the space and time discretization steps are linked up), and the refinement principle that adjusts the passage to grids more and more thin. The efficiency of the algorithms is proved.
For the entire collection see [Zbl 0919.00032].

MSC:

91A23 Differential games (aspects of game theory)
49N70 Differential games and control
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