Differential games of pursuit with the quality functional and their application to problems of dynamic resource allocation
The new classes of differential games pursuit-evasion with the phase constraints, the terminal set and functionality, the fixed time of the game end on the Cartesian product of the set of closed subsets of the Euclidean space and set of continuous functions are solved. With the help of new differential-operator constructions, a general approach to the construction of optimal strategies of each of the players is invented. The new properties of H-convex and matrix-convex sets, matrix-convex and quasi-convex functionals are obtained, the new methods of constructing the optimal counterstrategies of pursuer for the general classes of linear differential games are developed. Semigroups of operators (which, in general, dependent on a pair of sets and functions) are built for differential games with the possible outputs of the trajectories of a dynamic system to the terminal set at unlimited time in
To describe the structure of differential games and for constructing the optimal strategies of the players a new mathematical operator apparatus of matrix-convex sets and quasi-convex functions was created and implemented. On the basis of the developed theory of semigroups of operators a new approach to building strategies of B.M. Pshenichniy for such problems is invented. The new effective methods for solving the problem approximation-evasion of a special type with non-fixed time of ending the game are developed. The new algorithms for finding the optimal counterstrategies are validated.
The new mathematical models and methods of more effective dynamic resource distribution using the theory of differential games with a terminal set and functionality are developed. The urgent problems of flows control in networks with incomplete information are solved. For the first time the optimization problem of flows traffic is solved, and its solution satisfies the generalized Kirchhoff's laws. The results are applied to the classes of problems of a constructive distribution of hydro resources in irrigation systems of the network structure. The cases of the mixture support (water, gas deposits from various sources) with different quality are considered separately. The new mathematical models, methods and algorithms of solving tasks of delivery of the mixture of substance along the network in certain proportions or with specified quantitative limitations are created. The obtained generalizations for the first time allowed to solve the problem of calculating the diverse global distribution networks and pipelines.