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).
Figure 1. The sheaf solutions of CS (2.1) in two admissible controls.
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):
with
if 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
Figure 2. The sheaf of SAM shooting down aircraft noise bomb attack as B52.
(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.
NOTES