Singular perturbations in optimal control problems

Abstract

One presents a survey of the works on the theory of optimal control by deterministic objects, described by systems of ordinary differential or difference equations, where the investigation is carried out with the aid of the methods of the theory of singular perturbations. One analyzes the possibility of the application of the theory of singular perturbations to the investigation of control problems with large amplification coefficient in the feedback circuit, to the description of sliding regimes in systems with variable structure, and to the construction of effective numerical algorithms for the solution of optimization problems. INTRODUCTIONIn the last decades the mathematical theory of optimal processes has been intensively developed. Considerable attention has been given to problems with small perturbations. The application of the methods of perturbation theory (asymptotic methods) in problems of optimal control is useful because of the following reasons.i. In mathematical modeling one neglects frequently various small quantities and as a result one operates with idealized models~ The theory of perturbations allows us to establish a correspondence between the exact (perturbed) and the simplified (unperturbed, degenerate) models. This correspondence is established by the investigation of a limiting process from the solution of the perturbed problem to the solution of the unperturbed one. The theory of perturbations allows us to obtain corrections to the solution of the simplified problem (asymptotic expansion) and allows us to study the sensitivity of the solution of the exact problem with respect to small perturbations.2. The asymptotic theory gives us the possibility to obtain a qualitative picture of the solution~ 3. The knowledge of the as~cmptotics allows us to offer economical computational procedures for the approximate solution of the initial problems of optimal control, the solution of which is difficult or practically impossible because of nonlinearity, computational instability (as a consequence of the "rigidity"), high dimensionality, etco 4. Small perturbations in optimal control problems can be introduced artifically and then the theory of perturbations emerges as a method of investigation of the initial, in a certain sense "bad" (for example, ill-posed) problem.The investigation of optimal control problems by the methods of perturbation theory presents interest also for perturbation theory itself since its development is stimulated by the various new problems which emerge. The survey of F. L. Chernous'ko and V. B. Kolmanovskii The present survey is devoted to deterministic objects described by systems of ordinary differential or difference equations. Stochastic objects and objects described by partial differential equations are virtually not considered in this survey. The given bibliography does not have the pretention of absolute completeness. Additional references can be found in the works cited in the survey. We give some definitions and notations which will be used in the survey. The symbol ...

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