Dmitry Karamzin

On some extension of optimal control theory

Abstract Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.

On the extension of classical calculus of variations and optimal control to problems with discontinuous trajectories

In this article, we present a proper extension of the problems of classical calculus of variations and of optimal control to the case of discontinuous trajectories. An appropriate impulsive control problem is introduced as a result of such extension. Existence theorems are proved under assumptions of the Filippov type.

Comments on Paper “On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints”

This Forum Note concerns the question of necessary optimality conditions in optimal control problems subject to state constraints. Some critical remarks about a recently published paper are made.

On a Few Questions Regarding the Study of State-Constrained Problems in Optimal Control

The article is focused on the investigation of the necessary optimality conditions in the form of Pontryagin’s maximum principle for optimal control problems with state constraints. A number of results on this topic, which refine the existing ones, are presented. These results concern the nondegenerate maximum principle under weakened controllability assumptions and also the continuity of the measure Lagrange multiplier.

Minimax optimal control problem with state constraints

Abstract In this article, nondegenerate necessary conditions of optimality are derived and discussed for the so-called “minimax” optimal control problems with state constraints. In this class of problems, the data depends on an unknown parameter that takes values in a given compact set called “uncertainty set”. The solution of the problem is considered for the worst case value of the unknown parameter. The result is obtained in the two stages: after the basic necessary conditions of optimality are derived, additional conditions ensuring their nondegeneracy in the minimax context are obtained.

On the Study of Complex Impulsive Systems

The article investigates complex impulsive systems in which the so-called controlling systems jumps effect emerges. In particular, this research includes the correctness of the solution to the impulsive control system and approximation lemmas. A 3D model example is provided which illustrates the relevance of the considered approach to the study of complex impulsive systems.

On properness of impulsive extension

This article investigates properness of an impulsive extension of the conventional optimal control problem. This includes both the correctness of the solution to impulsive control systems arising as result of an impulsive extension of ordinary differential systems, and existence theorems. Correctness in the classic Cauchy sense is proved. New approximation lemmas are introduced proved that guarantee the sensitivity to small perturbations in control variables. Filippov-type existence theorems are established.

On the extensions of optimal control theory

We consider an impulsive control problem with state constraints which arises as result of extension of the classic theory. Filippovs type existence theorem is proved.