Mathematical Modelling and Applications
Volume 1, Issue 1, October 2016, Pages: 1-7

Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints

Mohammed Shehata

Department of Mathematics, Faculty of Science, Jazan University, Kingdom of Saudi Arabia

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To cite this article:

Mohammed Shehata. Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints. Mathematical Modelling and Applications. Vol. 1, No. 1, 2016, pp. 1-7. doi: 10.11648/

Received: September 6, 2015; Accepted: September 18, 2015; Published: October 12, 2016

Abstract: In this paper, the minimum time problem for differential systems of parabolic type with distributed control and control - state constraints are considered. The minimum time problem is replaced by an equivalent one with fixed time and the necessary optimality conditions of time-optimal control are obtained by using the generalized Dubovitskii-Milyutin Theorem (see [1]).

Keywords: Time-Optimal Control Problem, Parabolic System, Dubovitskii - Milyutin Method, Canonical Approximations, Optimality Conditions

1. Introduction

The most widely studies of the problems in the mathematical theory of control are the " " control problems. The simple version, is the following optimization problem :


where ,  are spaces of admissible control and states respectively. In order to explain the results we have in mind, it is convenient to consider the abstract form of the Dubovitskii-Milyutin theorem. At the first we recall some definitions of conical approximations [2], [3] and cones of the same sense or of the opposite sense [3]. Let  be a set contained in a Banach space  and  be a given functional.

Definition 1 A set

where  as  is called tangent cone to the set  at the point

Definition 2 A set

 where  is a neighborhood of , is called the admissible cone to the set  at the point

Definition 3 A set

 is called the cone of decreases of the functional  at the point

All the cones defined above are cones with vertices at the origin. The cones ,, are open while the , is closed. If , then  does not exist. Moreover, if   then

If the cones , and  are convex, then they are called regular cones and we denote them by , and respectively.

Let   be a system of cones and   be a ball with center  and radius  in the space

Definition 4 The cones   are of the same sense if    so that  we have  ( or equivalently the inequality  implies the inequality  , ).

Definition 5  The cones   are of the opposite sense if    so that

Remark 1  Form definitions 4 and 5 it follows that the set of cones of the same sense is disjoint with the set of cones of the opposite sense. If a certain subsystem of cones is of the opposite sense, then the whole system is also of the opposite sense.

Definition 6 Let  be a cone in . The adjoint cone  of  is defined as

where  denotes the dual space of

Let  be Banach space,   represent inequality constraints,    represent equality constraints and    is given functional .

Theorem 1 ( [1] )Assume that :

(i)        (i)  is convex and continuous,

(ii)      the cones   are convex,

(iii)     the cones  are either of the same sense or of the opposite sense, then  is a solution of the problem

if and only if the following equation ( Euler -Lagrange equation )must hold:

where ,

and  with not all functionals equal to zero simultaneously.

The above generalization of the Dubovitskii- Milyutin theorem is based on the definitions of the regular cones RTC, RFC, RAC and cones of the same sense and of the opposite sense which are introduced above. But for the purpose of our problems we are going to use the following sufficient condition for two cones to be of the same sense .

Theorem 2 ( [3]) Let   be a cone of the form    where  is a cone in  (- normed spaces ). If the operator M is linear and continuous, then

and the cones ,  are of the same sense.

Various optimization problems associated with the optimal control of distributed parameter systems have been studied in [6]-[7],[10]-[13]. The problem of time-optimal control associated with the parabolic systems have been discussed in some papers.  In [6]  the existence of a time-optimal control of system governed by a parabolic equation has been discussed. In [5], the maximum principles for the time optimal control for parabolic equation is given. All these results concerned the time optimal control problems of systems governed by only one parabolic equation and only control constraints.  In [14]-[25], the above results for systems governed by one parabolic equation are extended to the case of  co-operative parabolic or hyperbolic systems with only distributed control constraints. In the present paper, we will consider time-optimal distributed control problem for the following  co-operative linear parabolic system with control-state constraints (here and everywhere below the index ):


where  is a bounded open domain with smooth boundary    is a given functions,  represents a distributed control and  ( ) are a family of  continuous matrix operators,

with co-operative coefficient functions  satisfying the following conditions:


2. Solutions of Co-operative Parabolic Systems

This section is devoted to the analysis of the existence and uniqueness of solutions of system (1). Let  be the usual Sobolev space( see [4]) of order  one which consists of all  whose distributional derivatives   and  with the scalar product


We have the following dense embedding form ( see [4]) :

where  is the dual of

For  and , let us define a family of continuous bilinear forms



Lemma 1 If  is a regular bounded domain in  with boundary  and if  is positive on  and smooth enough ( in particular ) then the eigenvalue problem:

possesses an infinite sequence of positive eigenvalues:

Moreover is simple, its associate eigenfunction  is positive, and is characterized by:


Proof. See[5].

Now, let


Lemma 2 If (2) and (5) hold then, the bilinear form (3) satisfy the Gårding inequality


Proof. In fact

By Cauchy Schwarz inequality and (4),we obtain

Finally, from (5) we have (6).

Under the above lemma (Lemma 2) and using the results of Lions [6] and Lions and Magenes [7] we can prove the following theorems:

Theorem 3 Assume that (2) and (5) hold. Then, problem (1) has a unique weak solution if  and   Moreover, the mapping   is continuous from   

3. Control Problem

Let us consider the following optimization problem


under the following constraints:



Notation 1 We will call the problem (7)-(9), problem I

The optimization problem I can be replaced by another equivalent one with a fixed time  To show that we need tow auxiliary lemmas.

Lemma 3 Let  be the optimal time for the problem I. If  then (boundary of)

for any set  satisfying (7)-(8)

Proof. Any solution of (8) is continuous with respect to. If is not true, then there exists an admissible state  such that the observation  Thus a  exists so that . This contradicts the optimality of

Lemma 4 Let  be the optimal time for the problem I, let  and  be an optimal control and corresponding state, respectively. Then there exist a non-trivial vector  so that the pair  is the optimal for the following control problem with the fixed time :


Proof. The linearity of the equations (8)  implies that the endpoints  of all admissible states  form a convex set  From Lemma 3 we have  and

Since  thus there exists a closed hyperplane separating  and containing , i.e. there is a nonzero vector  such as[8]

This completes the proof.

Remark 2 The method fails if , e.g. in the case when  consists of a single point.

Remark 3 If the set  has a special form i.e


where  and  are given, then  is Known explicitly and is expressed by

According to Lemma 4, problem I is equivalent to the one with the fixed time  and the performance index in the form (10).

Let us denote by  the sets in the space  as follows




Thus the optimization problem I may be formulated in such a form

 subject to   (15)

We approximate the sets  and  by the regular tangent cones (RTC),  by the regular admissible cone (RAC) and the performance functional by the regular cone of decrease (RFC).

The tangent cone to the set at  has the form


where  is the Fréchet differential of the operator

mapping from the space  in to the space where

 According to Theorem3 on the existence of solution to the equation (8) it is easy to prove that  is the mapping from the space on to the space  as required in the Lusternik Theorem([2]).

According to (13) the tangent cone  to the set  at  has the form


where  is the tangent cone to the set  at the point  From [2] it is known that tangent cones are closed.

Applying the same arguments as in Section 2.2 from [9] we can show that

We have to use Theorem 2, to show that  and  are of the same sense. (Note that we do not need to determine the explicit form of   in order to derive this conclusion.) It is enough to use the Theorem 3 about the existence and uniqueness of the solution for parabolic system (8)which determine   in (16). According to this theorem the solution of such a system depends continuously on the right side; i.e., in our case on so we can rewrite the cone given by (16) in the form


where   is a linear and continuous operator. Then, applying Theorem 2, to the cones given by (17) and (18), we get the assumption (iv) of Theorem 1 is satisfied.

The admissible cone  to the set  at  has the form


where,  is the admissible cone to the set  at the point .

Using Theorem 7.5 [2], the regular cone of decrease for the performance functional is given by


where  is the fréchet differential of the performance functional.

If  then the adjoint cone consists of the elements of the form(Theorem 10.2 in [2])


The functionals belonging to  have the form (Theorem 10.1 [2])

The functionals  and can be expressed as follows


where  and  (Theorem 10.1 in [2]),  is the support functional to the set  at the point  and,   is the support functional to the set  at the point (Theorem 10.5 [2]).

Since all assumptions of Theorem 1 are satisfied and we know suitable adjoint cones then we ready to write down the Euler -Lagrange Equation in the following form.


4. Special Case

Since  depended on the target set  we shall interpret (21) after choosing  in a less form fashion (11).

sNotation 2 We will call the problem I with   is given by (11), problem

In the present case, according to Remark 3  (23)

Introducing the adjoint variable  by the solution of the following systems

The existence of a unique solution for the above equation can be proved using Theorem 3 with an obvious change of variables)

Taking into account that  is the solution of  for any fixed , we obtain;


So,the Euler -Lagrange Equation (21) takes the form:


A number cannot be equal to 0 because in such a case all functionals in the Euler -Lagrange Equation would be zero which is impossible according to the DM Theorem. Using the definition of the support functional and dividing both members of the obtained inequalities by  from (22) we obtain the maximum conditions:

If , then the optimality conditions are fulfilled with equality in the maximum conditions. We have thus proved:

Theorem 4 Assuming that  is the optimal time for the problem  and and  are the optimal control and corresponding state respectively. Then,their exists the adjoint state  so that the following system of equations must be satisfied:

State equations;


Adjoint equations;


Maximum conditions;



5. Conclusion

In this study, we have derived the optimality conditions to a special co-operative parabolic systems with control-state conditions. Most of the results we described in this paper apply, without any change on the results, to more general parabolic systems involving the following second order operator:

with sufficiently smooth coefficients (in particular, ) and under the Legendre-Hadamard ellipticity condition

for all  and some constant

In this case, we replace the first eigenvalue of the Laplace operator by the first eigenvalue of the operator  (see [5]).


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