Lavrova O. Optimal control for systems of differential equations on time scales.

The thesis is concerned with investigation the problems of optimal control for systems of differential equations on time scales. Sufficient conditions of existence of optimal control for systems of differential equations on finite and infinite intervals of time scale to the moment of the exit the solution from some domain are obtained. An analogue of Pontryagin’s maximum principle for dynamic equations on time scales is obtained, which gives the necessary optimality conditions in terms of the Pontryagin function and the corresponding Lagrange multipliers. Sufficient conditions of optimality are obtained for linearly convex controlled problems in terms of the maximum principle. Dynamic programming method is extended on the optimal control problems on time scales. The general Bellman equation on the time scales is obtained (the Hamilton – Jacobi – Bellman equation). The conception of viscous solution for such equation was introduced. The conditions for the uniqueness of such solution for the corresponding boundary value problem are established. The relationship between the problem for optimal control on the time scale and the same problem for ordinary differential equations is investigated. The conditions of convergence of minimizing control sequences to the weakened optimal control are obtained.