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