** Advances in Pure Mathematics ** Vol. 3 No. 6 (2013) , Article ID: 37447 , 11 pages DOI:10.4236/apm.2013.36077

Optimality of Distributed Control for n ( n Hyperbolic Systems with an Infinite Number of Variables

Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, Saudi Arabia

Email: drahqamlo@yahoo.com

Copyright © 2013 Ahlam Hasan Qamlo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received June 10, 2013; revised July 13, 2013, accepted August 11, 2013

**Keywords:** Hyperbolic Operators; Cooperative Systems; Spaces With an Infinite Number of Variables; Existence and Uniqueness of Solutions; Distributed Control Problems; Control Constraints; Variable Coefficients

ABSTRACT

In this paper, we study the existence of solutions for 2l order (n ´ n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients. The necessary and sufficient conditions for optimality of the distributed control with constraints are obtained and the set of inequalities that defining the optimal control of these systems are also obtained.

1. Introduction

The optimality conditions for systems consisting of only one equation and for n × n systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables have been discussed for example in [1-11].

In addition, optimal control problems for systems involving operators with an infinite number of variables for non-standard functional and time delay have been introduced in [12,13].

Furthermore, time-optimal control of systems with an infinite number of variables has been studied in [14-18].

Some applications of optimal control problem for systems involving Schrodinger operators are introduced for example in [19-21].

Making use of the theory of Lions [22] and Berezanskiĭ [23], we consider the optimal control problem of distributed type for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients. We first prove the existence and uniqueness of the state for these systems, then we find the set of equations and inequalities that characterize the optimal control of these systems. Finally, we impose some constraints on the control. Necessary and sufficient conditions for optimality with control constraints are derived.

This paper is organized as follows. In Section 1, we introduce spaces of functions of an infinite number of variables. In Section 2, we discuss the distributed control problem for these systems with Dirichlet conditions. In Section 3, we consider the problem with Neumann conditions.

2. Sobolev Spaces with an Infinite Number of Variables

This section covers the basic notations, definitions, and properties, which are necessary to present this work [24]. Let be a sequence of continuous positive probability weights such that

with respect to it we introduce on the region , the measure dr(x) by:

On we construct the space with respect to this measure such that is the space of all square integrable functions on i.e.

.

We shall set.

is a Hilbert space for the scalar product

associated to the above norm.

We consider a Sobolev space in the case of an unbounded region. For functions which are continuously differentiable l times up to the boundary G of and which vanish in a neighborhood of ¥, we introduce the scalar product

where is defined by

and the differentiation is taken in the sense of generalized function on, and after the completion, we obtain the Sobolev space, which is a Hilbert space and dense in. The space forms a positive space. We can construct the negative space with respect to the zero space and then we have the following imbedding

,

.

Analogous to the above chain we have a chain of the form

,

where

with the scalar product

and is its dual.

denotes the space of measurable function t ® f(t) on open interval (0,T) for the Lebesgue measure dt and such that

endowed with the scalar product

which is a Hilbert space.

Analogously, we can define the spaces

andthen we have a chain in the form

where with boundary.

By the Cartesian product, it is easy to construct the following Sobolev spaces with the norm defined by

where is a vector function and, also we can construct the Cartesian product for the above Hilbert spaces. Finally we have the following chain:

where

and

are the dual spaces of and resp.

3. Dirichlet Problem for 2l Order (n × n) Cooperative Hyperbolic System with an Infinite Number of Variables and with Variable Coefficients

In this section, we study the existence and uniqueness of solutions for 2l order cooperative systems governed by Dirichlet problems involving hyperbolic operators with an infinite number of variables and with variable coefficients, then we find the necessary and sufficient conditions of the optimal control of distributed type.

For, we have the following system:

(1)

where,

is a given function, and are bounded functions such that

(2)

(3)

System (1) is called cooperative if (2) holds.

The operator in system (1) is 2l order hyperbolic operator with an infinite number of variables with

[23] is given by:

(4)

since q(x,t) is a real valued function in x which is bounded and measurable on, such that

(5)

Definition 1:

For each t Î (0,T), we define a bilinear form

by

where

Then,

,

(6)

3.1. The Existence and Uniqueness of Solution

Lemma 1:

The bilinear form (6) is coercive on, that is, there exists c, c_{1}_{ }Î R, such that:

(7)

Proof:

We have,

thus,

From (2), (3), and (5), we deduce

then,

then,

since 0 < c £ 1, we have,

which proves the coerciveness condition on.

Under all the a bove consideration, theorems of Lions [22] and using the Lax-Milgram lemma we have proved the following theorem.

Theorem 1:

Under the hypotheses (2), (3) and (7), if

, and are given in

, and resp., then there exists a unique solution

for system (1).

Proof:

Let be a continuous linear form defined on by

then by Lax-Milgram lemma, there exists a unique element such that

(9)

Now, let us multiply both sides of first equation of system (1) by, then integration over Q, we have:

by applying Green’s formula

by entering the summation on the both sides, we have

by comparing the summation with (6), (8) and (9) we obtain:

then we deduce that:

which completes the proof.

3.2. Formulation of Dirichlet Problem

The space being the space of controls. For a control, the state

of system (1) is given by the solution of

(10)

.

The observation equation is given by

.

N is given as

such that,

.

For a given, the cost function is given by

. (11)

4. Control Constraints

The set of admissible controls U_{ad} is a closed convex subset of, Then the control problem is to find inf J(v) over U_{ad}.

Then using the general theory of Lions [22], there exists a unique optimal control u Î U_{ad} such that J(u) = inf J(v) for all v Î U_{ad}. Moreover, we have the following theorem which gives the necessary and sufficient conditions of optimality.

Theorem 2:

Assume that (7) holds and the cost function is given by (11). The necessary and sufficient conditions for

to be an optimal control are the following equations and inequalities:

(12)

with

,

,

(13)

together with (10), where

is the adjoint state.

Proof:

The optimal control is characterized by [23]:

that is

this inequality can be written as

(14)

Now, since

by using Green’s formula, (3) and (10), we have

Then

and

(15)

Since the adjoint equation for hyperbolic systems in Lions [22] takes the following form:

then, from (15) we obtain the first equation in (12), and from theorem1, system (12) admits a unique solution which satisfies

.

Now, we transform (14) by using (12) as follows:

using Green’s formula, (10) and (12), we obtain

using (10), we have

which is equivalent to.

Thus the proof is complete.

5. Neumann Problem for 2l Order (n × n) Cooperative Hyperbolic System with an Infinite Number of Variables and with Variable Coefficients

In this section, we discuss the optimal control for 2l order cooperative non-homogenous Neumann systems involving hyperbolic operators with an infinite number of variables and with variable coefficients.

(16)

where, for all 1 £ i £ n,

is a given function in

and the operator in system (16) is 2l order hyperbolic operator with an infinite number of variables with

is given by:

since q(x,t) is defined as in (5).

For each t Î (0,T), we define a bilinear form

as in (6).

5.1. The Existence and Uniqueness of Solution

Lemma 2:

The bilinear form is also coercive on, that is, there exists c, c_{1}Î R, such that:

(17)

Proof:

Since is everywhere dense in with topological inclusion, then we have

. (18)

By using (18) in (7), we obtain

which proves the coerciveness condition on.

By the Lax-Milgram lemma, we shall introduce the following theorem which gives the existence and uniqueness of the state for system (16).

Theorem 3:

Under the hypotheses (2), (3) and (17), if

,

and are given in,

and resp., then there exists a unique solution

for system (16).

Proof:

Let be a continuous linear form defined on by

(19)

then by Lax-Milgram lemma, there exists a unique element

such that (9) is satisfied.

by applying Green^{’}s formula

by entering the summation on the both sides, we have

by comparing the summation with (6), (8) and (9) we obtain:

then we deduce that:

which completes the proof.

5.2. Formulation of Neumann Problem

The space is the space of controls. The state

of system (16) is given by the solution of

(20)

.

The observation equation is given by

.

For a given, the cost function is given by

(21)

where M is a positive constant.

The control problem then is to find inf J(v) over U_{ad} with the same control constraints in Section II.

Then as in Section II, there exists a unique optimal control uÎ U_{ad} such that

. (22)

Under the given considerations, we may apply theorems of Lions [22] as in Section II to obtain the following theorem:

Theorem 4:

The necessary and sufficient conditions for optimality of the control problem (20), (21) and (22) are given by the following equations and inequalities:

(23)

with

,

,

together with (16).

The case of no constraints on the control:

In the case of no constraints on the control, i.e., the condition (13) reduces to

hence.

Example 1:

If we take n = 2 in Dirichlet problem (1) with the same conditions of coefficients (2) and (3), then the space of controls is. For a control

the state

of the system is given by the solution of

(24)

.

The necessary and sufficient conditions for the optimality are the following equations and inequalities:

(25)

,

(26)

together with (24), where is the adjoint state.

Example 2:

If we take

. (27)

Thus there are no constraints on then the inequality (26) is equivalent to

(28)

Thus the optimal control is given by the solution of the following set of equations and inequalities

(29)

Further

(30)

6. Conclusions

The main result of this paper finds the necessary and sufficient conditions of optimality of distributed control for 2l order (n ´ n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients that give the characterization of optimal control (Theorem 2, 4).

Also it is evident that by modifying:

• the boundary conditions (Dirichlet, Neumann, mixed)

• the nature of the control (distributed, boundary)

• the nature of the observation (distributed, boundary)

• the initial differential system

• the number of variables

• the type of equation (elliptic, parabolic and hyperbolic)

• the type of coefficients (constant, variable)

• the type of system (non-cooperative, cooperative)

• the order of equationmany of variations on the above problems are possible to study with the help of Lions formalism.

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