A. A. Chikrii

Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order

A general method for solving game problems of approach for dynamic Volterra-evolution systems is presented. This method is based on the method of resolving functions [5] and the techniques of the theory of multivalued mappings. Properties of resolving functions are studied in more detail. Cases are separated where resolving functions can be derived in an analytical form. The scheme proposed covers a wide range of functional-differential systems, in particular, integral, integro-differential, and differential-difference systems of equations that describe the dynamics of a conflict controlled process. Game problems for systems with fractional Riemann-Liouville derivatives and regularized Dzhrbashyan-Nersesyan derivatives are studied in more detail. We will call them fractal games. An important role in the presentation of solutions of such systems is played by the generalized Mittag-Leffler matrix functions, which are introduced here. The use of asymptotic representations of these functions within the framework of the scheme of the method allows us to establish sufficient conditions of resolvability of game problems. A formal definition of parallel approach is given and illustrated by game problems for systems with fractional derivatives.

Control Game Problems for Quasilinear Systems with Riemann-Liouville Fractional Derivatives

Conflict-controlled processes for systems with Riemann-Liouville fractional derivatives of arbitrary order are studied. The solutions to such systems are presented in the form of an analog of the Cauchy formula. Sufficient conditions for completion of the game are obtained using the method of resolving functions. These conditions are based on the modified Pontrjagin condition expressed in terms of the Mittag-Leffler matrix functions. To find them, the Lagrange-Sylvester interpolating polynomials are applied.

Analytical Method for Solution of the Game Problem of Soft Landing for Moving Objects

A game problem of pursuit of a controlled object moving in a horizontal plane, by another object, moving in a three-dimensional space, is treated. The dynamics of the players models motion in a medium characterized by friction. Initial phase states are described, and sufficient conditions on parameters of a conflict-controlled object are derived, for which the soft landing may be performed.

The Scientific Heritage of B. N. Pshenichnyi

Pshenichnyis achievements in the investigation of necessary extremum conditions, computational methods of optimization, and differential games are outlined at a qualitative level.