Pontryagin’s maximum principle for dynamic systems on time scales

In this work, an analogue of Pontryagin’s maximum principle for dynamic equations on time scales is given, combining the continuous and the discrete Pontryagin maximum principles and extending them to other cases ‘in between’. We generalize known results to the case when a certain set of admissible values of the control is not necessarily closed (but convex) and the attainable set is not necessarily convex. At the same time, we impose an additional condition on the graininess of the time scale. For linear systems, sufficient conditions in the form of the maximum principle are obtained.     

A constructive method for finding periodic solutions of differential systems

Coefficient conditions for the existence of periodic solutions of degenerate systems of differential equations are obtained. Iterative algorithms for finding these solutions are developed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 468–476, April, 1995.     

Invariant Sets of Impulsive Systems and Their Stability

For impulsive systems, in terms of Lyapunov functions, we obtain conditions for the existence of invariant sets and study the stability of these sets.     

Analytic representation of periodic solutions of linear differential systems

We study the problem of periodic solutions of linear differential systems with small parameter. We establish new conditions for the existence and uniqueness of periodic solutions of these systems, which can be efficiently verified. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 731–735, May, 1997.