Kichmarenko O. Asymptotic analysis of optimal control problems for differential and functional-differential systems.

Three types of dynamically controlled objects are considered in the dissertation: systems of ordinary differential equations, systems of functional-differential equations, and some classes of difference equations. The problems of optimal control of these systems on a finite time interval and on an infinite horizon are investigated using asymptotic methods. For functional-differential system with distributed control on a finite interval the theorems of existence and uniqueness of the solution of the initial problem are proved. It is some analogue of the Carath\'{e}odory theorem for ordinary differential equations. The conditions of extension of the solution of the initial problem to the boundary of the domain is established. It is shown that the solution of the original problem cannot go to infinity in finite time. For the first time, sufficient conditions are obtained for the existence of optimal control for a system of functional differential equations which is nonlinear with respect to the phase variables and linear with respect to the control with an unfixed moment of reaching the domain boundary for a problem on a finite interval in terms of the right-hand sides and a quality criterion, which makes them convenient for verification. In the case when the domain in the statement of the problem is unbounded the theorem, which guarantees existence of optimal control in this case, is proved. The sufficient coefficient conditions for existence of optimal control for a system of functional-differential equations for the problem on the semi-axis also in terms of the right-hand side and the quality criterion are obtained. The relation between the solutions of the control problem on infinite interval and the same problem on finite interval is studied and results about these relations are proven.