Some Kinds of Sheaf Control Problems for Control Systems

Recently, the field of differential equations has been studying in a very abstract method. Instead of considering the behaviour of one solution of a differential equation, one studies its sheaf-solutions in many kinds of properties, for example, the problems of existence, comparison, ... of sheaf solutions. In this paper we study some of the problems of controllability for sheaf solutions of control systems.


Introduction
In [1][2][3][4] the authors have investigated sheaf solutions of control differential equations in the fields: comparison of sheaf solutions in the cases' two admissible controls and u t , and some initial conditions 0 0 x H  , 0 0 x H  , where the Hausdorff distance between the sets of initials 0 H and 0 H is enough small.The problems of sheaf controllability and sheaf optimization are still open.The present paper is organized as follows.In Section 2, we review some facts about sheaf solutions.In Section 3 we give many kinds of sheaf control problems, of sheaf controllability optimal problems.

Preliminaries
In n-dimension Euclidian space usually we have considered the control systems (CS): H is a collection of some given initials.
that will be controlable.In [2] the authors have compared the sheaf solutions for set control differential equations (SCDEs).
In [4] the author has study the problems (GC), (GA) and (GAZ) for set control differential equations (SCDEs).

   
Definition 2.5.A cut-set (a cross-area) of sheaf solution t u H  at time is denoted by: (2.5)

Main Results
Let's consider again the control systems (CS): where , Q is a compact set in and -admissible controls.Assume that for CS (3.1) there exists solution (2.2) and sheaf solution (2.4).We will need the following hypotheses on the data of control problem for CS (3.1): where .
we have two forms of sheaf solutions: where      0 0 x t x t x t u t     -solution of CS (2.1) (see Figure 1).
The pair of the sets will be controllable if: 1)   x t belongs to solutions of CS (3.1), and As in results, we have one map moving 0    Definition 3.3.The control system (3.1) is said to be: We have Proof.Suppose that for CS (3.1) the right hand side We have satisfies condition of lemma 3.1 then for all there exists such that: Proof.Because solution of CS (3.1) is equivalent: by lemma 3.1 we have: such that: and satisfy the followings: then for any admissible controls     u t u t  we have: Proof.Beside (2.4) for 0 H and 0 H we have:

H x t x t u t x H u t U t I H x t x t u t x H u t U t I
 as following:  , we have: We say that for control system (3.1) are given OCP-the optimization control problem if it denotes: to Hamillton Jacobi Bellman (HJB)-partial differential equation: We have to find the optimal control   and there exists feedback such that: Proof.Assume that   x t  -one of solutions of control systems (3.1) such that there exists feedback :    Definition 3.5.We say that for control system (3.1) are given SOCP-the sheaf-optimization control problem if it denotes: where such that is solution to (HJB)-partial differential equation: ) is a solution of HJB partial differential Equation (3.10) with the boundary conditions: and u(t) is admissible control then for optimization control problem SOCP (3.10) there exists estimate: where , where   for the pu of shooting down aircraft noise bomb attack as B52 shot ckets can not su wil When using missiles not e if only 01 or 02 ro cceed.The rockets theit fire it will be the interference or escort aircraft l

Figure 1 .
Figure 1.The sheaf solutions of CS (2.1) in two admissible controls.

Figure 2 .
Figure 2. The sheaf of SAM shooting down aircraft noise bomb attack as B52.

Definition 3.1. The Hausdorff distance between set
The pair of the any sets 0 1 sets of sheaf-solutions of CS (3.1) satisfy an estimate: d