Some results on sheaf-solutions of sheaf set control problems

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Abstract

In this paper, we study the existence and some properties of solutions of the so-called set control differential equations (SCDE) and sheaf-solutions of sheaf set control problems.

Introduction

Recently, the study of set differential equations (SDE) in a semilinear metric space has gained much attention. Some initial and important results in this direction have been obtained in a series of works of Prof. V. Lakshmikantham and other authors. See [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. In [15], we considered the so-called set control differential equation (SCDE), then studied some existence results for solutions of SCDE. The concept of a sheaf-solution of a classical control problem was introduced in [12]. Instead of studying each solution, one studies the sheaf-solution, that means, a set of solutions. In this paper, we present some comparison results for SCDE and sheaf-solutions of sheaf set control problems concerning the SCDE on variation of controls. The paper is organized as follows. Section 2 reviews some concepts of Hausdorff distance, Hukuhara derivative, set differential equations. The sheaf set control problems and some new results on the dependence of sheaf-solutions on variation of controls are presented in Section 3.

Section snippets

Preliminaries

Firstly, we recall some notation and concepts which were presented in detail in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].

Let Kc(Rn) denote the collection of all nonempty, compact, convex subsets of Rn. Let A,B be two nonempty bounded subsets of Rn. The Hausdorff distance between A and B is defined as D[A,B]=max{supa∈Ainfb∈B‖a−b‖,supb∈Binfa∈A‖a−b‖}. In particular D[A,θ]=‖A‖=sup{‖a‖:a∈A}, where θ is the zero element of Rn which is regarded as a one point set.

The Hausdorff metric

Main results

We consider the initial valued problem (IVP) for a set control differential equation as follows: DHX=F(t,X(t),U(t)),X(t0)=X0, where F∈C[I×Kc(Rn)×Kc(Rp),Kc(Rn)],t∈I⊂R+, state X(t)∈Kc(Rn) and the control U(t)∈Kc(Rp).

If U:I→Kc(Rp) is integrable, then it is called an admissible control. Let U be a set of all admissible controls. The mapping X∈C1[I,Kc(Rn)] is said to be a solution of (3.1) on I if it satisfies (3.1) on I. Since X(t) is continuously differentiable, we have X(t)=X0+∫t0tDHX(s)ds,t∈I.

Acknowledgments

The authors gratefully acknowledge the anonymous referee for the valuable remarks and careful reading which improved the presentation of this paper. The authors wish to express their special thanks to Prof. V. Lakshmikantham and the other authors for their results on set differential equations in [1], [2], [3], [5], [6], [9], [10], [11] which were adapted to set control differential equations in this paper.

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