Optimization of game interaction of fractional-order controlled systems

The article concerns game problems for controlled systems with arbitrary Riemann-Liouville fractional derivatives [S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993.]. Under fixed controls of the players, solution of such systems is presented in the form of analog of Cauchy formula. On the basis of the method of resolving functions [A.A. Chikrii, Conflict-Controlled Processes, Kluwer Academic Publ, Boston, London, Dordrecht, 1997.], we derive sufficient conditions for the finite-time game termination from given initial states. At the heart of these conditions lies the modified Pontryagin's condition [A.A. Chikrii, Conflict-Controlled Processes, Kluwer Academic Publ, Boston, London, Dordrecht, 1997; L.S. Pontryagin, Selected Scientific Works, Vol. 2, Nauka Moscow, 1988 (in Russian).] consisting in the nonemptiness of certain set-valued mappings. These mappings are expressed through the control domains of the players and the generalized Mittag-Leffler functions for the matrix of the system. To evaluate the latter, the apparatus of interpolating polynomials of Lagrange-Sylvester is used [F.R. Gantmakher, Theory of Matrices, Nauka, Moscow, 1967 (in Russian).]. The results are illustrated in the example with 'simple' matrix and fractional generalized controlling Pontryagin's example [L.S. Pontryagin, Selected Scientific Works, Vol. 2, Nauka Moscow, 1988 (in Russian).]. In so doing various specific cases are analyzed, including known model examples 'The Boy and the Crocodile' and 'Isotropic Rockets' [Isaacs, R., Differential Games, John Wiley, New York, 1965.].