A differential game of unlimited duration
The properties of the value function (VF) in a differential game of unlimited duration with depreciating performance functional are studied, and two methods of approximating the VF are compared. The VF does not satisfy a Lipschitz condition, due to the type of functional. It is therefore not possible to prove in the general case differential inequalities for the usual directional derivatives. To overcome this difficulty, a generalization of the directional derivative of a continuous function is proposed. It consists in ``smearing'' the direction by higher-order increments than the increment of the argument. Necessary and sufficient conditions on the VF are obtained in terms of the directional derivative and of the conjugate derivatives. It is shown that the differential inequalities used to find the viscous solutions and the inequalities of the present paper are equivalent in every position. It is also shown that the method of discrete approximation of the stationary Hamilton-Jacobi equation for control problems is likewise applicable for problems of differential game theory. This method is shown to be equivalent to the familar following procedure. Problems of the present type arise e.g., when modelling processes with depreciating performance factor. Such problems of optimal programmed control were previously studied by \textit{I. C. Dolcetta} [Appl. Math. Optimization 10, 367-377 (1983; Zbl 0582.49019) and ibid. 11, 161-181 (1984; Zbl 0553.49024)] and \textit{P.-L. Lions} and \textit{P. E. Souganidis} [SIAM J. Control Optimization 23, 566-583 (1985; Zbl 0569.49019)] in the absence of unmonitored noise, which has to be regarded as an opponent player.
scientific article; zbMATH DE number 4087469
scientific article