Abstract
We discuss the calculation of discriminating kernel for the discrete-time dynamic game and continuous-time dynamic game (namely differential game) using the viability kernel and reachable set. For the discrete-time dynamic game, we give an approximation of the viability kernel by the maximal reachable set. Then, based on the relationship between viability and discriminating kernels, we propose an algorithm of the discriminating kernel. For the differential game, we compute an underapproximation of the viability kernel by the backward reachable set from a closed target. Then, we put forward an algorithm of the discriminating kernel using the relationship of the discriminating and viability kernels. This means that the victory domain can be computed because it is computed by the discriminating kernel. The novelty is that we give two algorithms of the discriminating kernel for a dynamic game that contains two control variables, not one control variable as in differential inclusion.
Highlights
As an important part of control theory, game theory pours attention into economics, social, political science, and other behavioral sciences
Dynamic games are closely related to optimal control problems, there is a little difference between the two: there is a single control input u(t) and a single criterion to be optimized in an optimal control problem, and dynamic game theory generalizes this to two control inputs u(t), v(t) and two criteria
Using set-valued analysis and viability theory, we computed an approximation of the viability kernel by the maximal reachable set
Summary
As an important part of control theory, game theory pours attention into economics, social, political science, and other behavioral sciences. In [ ], quantitative and qualitative differential game problems are discussed using set-valued analysis and viability theory. In [ ], a method to compute overapproximations of the reachable set for nonlinear dynamic systems using trajectory piecewise linearized models is proposed. Reachability analysis and viability theory provide solid frameworks for control system of constrained dynamical systems in a set-valued fashion [ , ]. In [ ], an algorithm for computing the set of reachable states of a continuous dynamic game is discussed. An approximation of the viability kernel for the discrete dynamic game is computed by the maximal reachable set from a closed target. An underapproximation of the viability kernel for the differential game is computed by the backward reachable set from a closed target. In Section , we study the discriminating kernel of a continuous differential game
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