Minimum principle and controllability for multiparameter discrete inclusions via derived cones

Aurelian Cernea

doi:10.1155/ddns/2006/96505

Abstract

We consider a multiparameter discrete inclusion and we prove that the reachable set of a certain variational multiparameter discrete inclusion is a derived cone in the sense of Hestenes to the reachable set of the discrete inclusion. This result allows to obtain sufficient conditions for local controllability along a reference trajectory and a new proof of the minimum principle for an optimization problem given by a multiparameter discrete inclusion with endpoint constraints.

Highlights

The concept of a derived cone to an arbitrary subset of a normed space has been introduced by Hestenes in [8] and successfully used to obtain necessary optimality conditions in control theory
In our previous papers [3,4,5,6,7], we indentified certain derived cones to the reachable sets of “ordinary” differential inclusions, hyperbolic differential inclusions, and some other classes of discrete inclusions in terms of the variational inclusion associated to the differential inclusion and to the discrete inclusion
These results allowed to obtain a simple proof of the maximum principle in optimal control and sufficient conditions for local controllability along a reference trajectory

Summary

Introduction

The concept of a derived cone to an arbitrary subset of a normed space has been introduced by Hestenes in [8] and successfully used to obtain necessary optimality conditions in control theory. In our previous papers [3,4,5,6,7], we indentified certain derived cones to the reachable sets of “ordinary” differential inclusions, hyperbolic differential inclusions, and some other classes of discrete inclusions in terms of the variational inclusion associated to the differential inclusion and to the discrete inclusion. These results allowed to obtain a simple proof of the maximum principle in optimal control and sufficient conditions for local controllability along a reference trajectory.

Multiparameter discrete inclusions

Preliminaries

The main result

Applications