Minimum Time Problem for Co-operative Parabolic System with Control-State Constraints
Department of Mathematics, Faculty of Science, Jazan University, Kingdom of Saudi Arabia
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/j.mma.20160101.11
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 ).
Keywords: Time-Optimal Control Problem, Parabolic System, Dubovitskii - Milyutin Method, Canonical Approximations, Optimality Conditions
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 ,  and cones of the same sense or of the opposite sense . 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 (  )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:
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 ( ) 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 -,-. The problem of time-optimal control associated with the parabolic systems have been discussed in some papers. In  the existence of a time-optimal control of system governed by a parabolic equation has been discussed. In , 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 -, 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 ) of order one which consists of all whose distributional derivatives and with the scalar product
We have the following dense embedding form ( see ) :
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:
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  and Lions and Magenes  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
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().
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  it is known that tangent cones are closed.
Applying the same arguments as in Section 2.2 from  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 , 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 )
The functionals belonging to have the form (Theorem 10.1 )
The functionals and can be expressed as follows
where and (Theorem 10.1 in ), is the support functional to the set at the point and, is the support functional to the set at the point (Theorem 10.5 ).
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:
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 ).