Stability and Optimality of Solutions to Differential Inclusions via Averaging Method

Abstract

The averaging method is one of the most used tools to study dynamical systems. With the development of the theory of differential inclusions, a respective generalization of the averaging method followed the steps outlined in the theory of differential equations. Presently, it has been successfully applied to a wide range of problems involving differential inclusions, simplifying the study of the systems under consideration. In this work, the main development trends and methods in the application of the averaging method to the study of stability and optimality of solutions to differential inclusions are surveyed. A detailed list of references is given and some examples of applications are presented.

Averaging for ordinary differential equations perturbed by a small parameter

“…We point out also that under the conditions (H1) and (H2) in Theorem 2.1, it is only possible to obtain unilateral approximations, that is, the approximation of solutions of problems (2.1) by those of the averaged problem (2.2). The converse approximation is, in general, false as showed in [3], page 356, Example 1. However, when the problem (2.2) has a unique solution, this solution is approximated by any one of the problem (2.1) as it is stated by the following interesting result which is a particular case of Theorem 2.1.…”

“…The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those often considered in the existing literature as in [2], [4], [6]- [9], for instance (see also the survey paper [3]). Indeed, in all the references cited above, it is assumed that the right-hand sides of the equations are uniformly bounded.…”

Tight Bounds for Averaging Multi-Frequency Differential Inclusions, Applied to Control Systems

“…The averaged inclusion is obtained by computing the time average of the set-valued mapping. As a time-independent inclusion it is amenable to analysis, and applications of averaging in stabilization and optimality can be found in Gama and Smirnov [7]. In this paper we focus on estimating the difference, in the Hausdorff distance, between the solution sets of both systems.…”

“…For a reference on the averaging method of ordinary differential equations, the reader is referred to the book of Sanders, Verhulst and Morduck [10] and to works of Artstein [1] and Bright [5,6] for a modern treaty and improved estimates on the error, the line of which we follow in this paper. For a reference to results in differential inclusions refer to the review papers Klymchuk, Plotnikov and Skripnik [8] and to [7].…”

A method for solving differential inclusions with fixed right end

“…It is also known, that many optimal control problems under natural assumptions may be reduced to differential inclusions [4]. Finding solutions for a differential inclusion is important for applications [6][7][8]. As a rule, it is possible to obtain an analytical solution of a differential inclusion only in special cases but in the other cases one has to use numerical methods for this purpose.…”