ABSTRACT

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.

1. Introduction

In [1-4] the authors have investigated sheaf solutions of control differential equations in the fields: comparison of sheaf solutions in the cases’ two admissible controls and, and some initial conditions, , where the Hausdorff distance between the sets of initials and 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.

2. Preliminaries

In n-dimension Euclidian space usually we have considered the control systems (CS):

(2.1)

where,

. A solution to CS (2.1) is represented by:

(2.2)

, , is a collection of some given initials.

Definition 2.1. We say that a control is admissible, if:

1) satisfies (2.2) for all;

2) is bounded by norm.

That means the functions are measurable (integrable) satisfiying almost everywhere on the relationships (2.1) and (2.2), then is called the trajectory of the CS (2.1) and is called the control. Therefore, we shall always understand a pair of functions interrelated by the relationship (2.1) and (2.2). It is clear that several controls can correspond to one trajectory and if CS (2.1) has a nonunique solution, then several trajectory can correspond to one control.

Definition 2.2. A state pair of solutions of control systems (2.1) will be a controllable if after time we shall find a control such that:

(2.3)

Definition 2.3. A control system (2.1) is said to be:

(GC) global controllable if every state pair of set solution.

(GA) global achievable if for every we have a state pair of solutions that is GC.

(GAZ) global achievable to zero if for every we have a state pair 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 2.4. A sheaf solution (or sheaf trajectory) is denoted by a number of solutions that make into sheaves (lung one on top of the other and often tied together) for all:

(2.4)

Definition 2.5. A cut-set (a cross-area) of sheaf solution at time is denoted by:

(2.5)

3. Main Results

Let’s consider again the control systems (CS):

(3.1)

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):

(Hf1):

(Hf2):

where.

Assume that at all, for two admissible controls we have two forms of sheaf solutions:

(3.2)

where—solution of CS (2.1) (see Figure 1).

Definition 3.1. The Hausdorff distance between set and is denoted by:

Definition 3.2. The pair of the any sets will be controllable if after time we shall find a control and one map such that:

(3.3)

Theorem 3.1. Under Hypothes (Hf1), let —is initial, any set. The pair of the sets will be controllable if:

1) belongs to solutions of CS (3.1), and 2) is cut-set of sheaf solution to (3.1), that means

.

Proof. If belongs to solutions of CS(3.1) then it is

.

For any we have a pair that is controllable, because, where —cut-set of sheaf solutions with

As in results, we have one map moving to that means.                 

Definition 3.3. The control system (3.1) is said to be:

(SC1) sheaf controllable in type 1, if for all, there exists and admissible controls that satisfy then

(3.4)

(SC2) sheaf controllable in type 2 for any admissible control, if for all, there exists such that the initials with then

(3.5)

(SC3) sheaf controllable in type 3, if for all, there exist, such that the initials with and for any admissible controls that satisfy then

(3.6)

Lemma 3.1. Under Hypothes (Hf1), for all, there exists if control system (3.1) with:

then two cut -sets of sheaf-solutions of CS (3.1) satisfy an estimate:

(3.7)

Proof. Suppose that for CS (3.1) the right hand side satisfies (Hf1) then there exists unique solution which satisfies (2.2).

If—sheaf solution of CS (3.1) then for admissible control we have the cut-sets at any times, that satisfy estimate (3.7):

and.

We have 

   

Theorem 3.2. Assume that, under Hypothes (Hf2), the admissible controls that satisfy , then CS (3.1) is sheaf-controllable SC1.

Proof. Suppose that for CS (3.1) the right hand side satisfies (Hf2) then there exists unique solution which satisfies (2.2).

If—sheaf solution of CS (3.1) then for admissible control we have the cut-sets at every times, that satisfy estimate (3.7): and.

We have

as results the CS (3.1) is sheaf-controllable SC1.  

Corollary 3.1. If CS (3.1) is SC1, the right hand side satisfies condition of lemma 3.1 then for all there exists such that:

Proof. Because solution of CS (3.1) is equivalent: then

by lemma 3.1 we have:          

Theorem 3.3. Under hypothes (Hf1), assume that the initials for all, there exists such that: then for any admissible control we have:

that means CS (3.1) is sheaf controllable CS2.

Proof. We have estimate

For all, such that and

choosing then we have. As results imply that CS (3.1) is SC2.               

Theorem 3.4. Under Hypothes (Hf2), assume that for all and satisfy the followings:

1)

2)then for any admissible controls we have:

that means CS (3.1) is sheaf controllable CS3.

Proof. Beside (2.4) for and we have:

and estimate as following:

Choosing, we have:

          

Definition 3.4. We say that for control system (3.1) are given OCP—the optimization control problem if it denotes:

(3.8)

where, such that V(T,x) is solution to Hamillton Jacobi Bellman (HJB)—partial differential equation:

(3.9)

We have to find the optimal control for OCP (3.8).

Lemma 3.1. In optimization control problems (3.8) if then

Proof. Putting

we have integral for all:

Because

impilies that

  

Theorem 3.5. Assume that OCP (3.8) has and there exists feedback such that:

then exists optimal control for OCP (3.8).

Proof. Assume that—one of solutions of control systems (3.1) such that there exists feedback:

By lemma 3.3 we have

such that—optimal control for OCP (3.8).                                      

Definition 3.5. We say that for control system (3.1) are given SOCP—the sheaf-optimization control problem if it denotes:

(3.10)

where—integral on and

such that is solution to (HJB)—partial differential equation: 

(3.11)

Lemma 3.2. Assume that V(t, x) is a solution of HJB partial differential equation (3.10) with the boundary conditions:

If function

and u(t) is admissible control then for optimization control problem SOCP (3.10) there exists estimate:

Proof. Putting

we have:

where

then

By (*) we have

and then (**) impilies that

 

Theorem 3.6. (Necessary Conditions)

Assume that SOCP (3.10) has solution, that means there exists optimal control such that

and is a solution of HJBpartial differential equation (3.11) then the necessary conditions for this SOCP (3.10) are:

1)

2), where

Proof. Suppose that a function SOCP (3.10) that means

. Because V(t, x)-solution of HJBpartial differential equation (3.11):

withif function satisfies:

that integrable on sheaf solutions.

By lemma 3.2, if is admissible control then for optimization control problem SOCP (3.10) there exists estimate:

Assume that for SOCP (3.10) has optimal control then for all, we have    

Theorem 3.7. (Sufficient Conditions)

Assume that any admissible control for SOCP (3.10) and is a solution of HJB-partial differential Equation (3.11) then the sufficient conditions for this SOCP (3.10) are:

1)

2)

3) there exists such that

Proof. There exists the other admissible control, such that for SOCP (3.10) we have

By condition (1) of theorem 3.6 we have a function

that integrable on sheaf solutions.

We find the function from equation:

with condition.

By condition (2) of this theorem:

and implies that—optimal control for SOCP (3.10).                             

Example 3.1. When using missiles not for the purpose of shooting down aircraft noise bomb attack as B52 shot if only 01 or 02 rockets can not succeed. The rockets theit fire it will be the interference or escort aircraft will be explosive.

A problem arises: What to do in order to shoot down aircraft noise when operating in the sky. To solve this problem, we must fire simultaneously from SAM sites from 03 or more results. The rockets have to be controlled from headquarters and shot to pick the exact point-B52.

Mathematical model for problem shooting attack aircraft noise control system with (3.10), the test bundle (2.4) and optimization problem are (SOCP) in the above with n = 3 (see Figure 2).

4. Conclusions

The Sheaf Optimization problem for Control Systems

(SOPCS) have a high practical significance, as the series of SAM to destroy B52 attack aircraft with fighter jamming, or laser beam to destroy targets, like the beams in materials research of Physical nuclear, etc, … This paper described some types of sheaf optimal problems. We can solve them by Pontryagin’s Principle, Lyapunov’s Energy Function or by the Hamilton’s Principle. In this paper we present the necessary and sufficient conditions for this problem by Hamilton’s Principle, namely by HJB equations.

In the near future, we will set the numerical calculations can be applied to a clearer and will study the different Optimization problems with some controls.

5. Acknowledgements

The authors gratefully acknowledge the referees for their careful reading and many valuable remarks which improved the presentation of the paper.

REFERENCES

NOTES

^*Corresponding author.