Time-Optimal Control Problem for n × n Co-Operative Parabolic Systems with Control in Initial Conditions

In this paper, time-optimal control problem for a liner n n  co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state   0 y u  , with control so that an observation , u   y t hits a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.


Introduction
The "time optimal" control problem is one of the most important problems in the field of control theory.The simple version is that steering the initial state in a Hilbert space 0 y H to hit a target set K H  in minimum time, with control subject to constraints .

  u U H  
In this paper, we will focus our attention on some special aspects of minimum time problems for co-operative parabolic system involving Laplace operator with control acts in the initial conditions.In order to explain the results we have in mind, it is convenient to consider the abstract form.
Let and V H be two real Hilbert spaces such that is a dense subspace of V .H Identifying the dual of H with , H we may consider , V H V   where the embedding is dense in the following space.Let be a family of continuous operators associated with a bilinear forms defined on which are satisfied with Gårding's inequality for   , 0, y V t T   and 0 Then, from [1] and [2], for have a unique weak solution such that y     0, ; .

y C T H 
We shall denote by   ; y t u the unique solution of the Equation (2) corresponding to the control u.The time optimal control problem that we shall concern reads: where K is a given subset of , H which is called the target set of the Problem (3) We call the number 0  as the optimal time for the time optimal control Problem (3).Three questions (problems) arise naturally in connection with this problem: In the works [1] and [3], the existence of time optimal controls of the following controlled linear parabolic equations with distributed control was obtained: where The results in [3] partly overlap with results in [1] and they were shown that if the system ( 5) is controllable and if then the corresponding time optimal control problem has at least one solution and it is bangbang.
In the work [4], the authors gave a sufficient and necessary condition for the existence of time optimal control for the problem with the target set   0 K  and certain controlled systems.These results will be stated as follows.Consider the following controlled system where is a real number.Let be the corresponding eigenfunctions, which forms an orthogonal basis of

L 
We take the target set to be the origin K   0 in and the control set to be the set where  is a positive number, namely, then the corresponding time optimal control problem has at least one solution if and only if 1 .
a   More early, in the works [5][6][7], the time optimal controls problem for globally controlled linear and semilinear parabolic equations was considered.
In our papers [8,9], the time optimal control problem of n n  co-operative hyperbolic systems with different cases of the observation and distributed or boundary controls constraints was considered.
In [10], optimal control of infinite order hyperbolic equation with control via initial conditions was considered.
In the present paper, the above results for the time optimal control of systems governed by parabolic equations are extended to the case of co-operative parabolic systems as well as control via initial conditions.First, the existence and uniqueness of solutions for n n  n n  co-operative parabolic system are proved under conditions on the coefficients stated by the principal eigenvalue of the Laplace eigenvalue problem, then the time optimal control problem is formulated and the existence of a time optimal control is proved.Then the necessary and sufficient conditions which the optimal controls must satisfy are derived in terms of the adjoint.Finally, the scaler case is given.

n × n Co-Operative Parabolic Systems
Let   1 0 , H  be the usual Sobolev space of order one which consists of all and 0    with the scalar product norm where .
H  Here and everywhere below the vectors are denoted by bold letters.For , let us define a family of continues bilinear forms ;.,.: by where , and , are positive functions in , 0 when and when The bilinear form (7) can be but in the operator form: ,

 
n is matrix operator which maps onto and takes the form with boundary and if is positive on ,  m  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 2.2.If (8) and (10) hold then, the bilinear form (7) satisfy the Gårding inequality , By Cauchy Schwarz inequality and (9), we obtain L  y to two sides, then we have the result..
We can now apply Theorem 1.1 and Theorem 1.2 Chapter 3 in [1] (with and ) to obtain the following theorem: 8) and (10) hold, then there exist a unique solution

Minimum Time and Controllability
We denote the unique solution of (11), at time for each control t  

 
, ; x t y u when the explicit dependence on x is required.We can now formulate the time optimal control problem corresponding to the n n  co- operative parabolic system (11): ; is the solution of 11 , , , : , , , : , and , 0  are given.

   
there exists a 0, and with ; Proof.Let us first remark that by translation we may always reduce the problem of controllability to the case were the system (11) with We can show quit easily that (11) is approximately controllable in if and only if, is dense in By the Hahn-Banach theorem, this will be the case if for all implies that Let us introduce the adjoint state   ; p t u by the solution of the following system The existence of a unique solution for the Problem (16) can be proved using Theorem 2.3, with an obvious change of variables.
Multiply the first equation in (16) by

 
; i y t u and integrate by parts from 0 to ,  we obtain the following identity: and so, if (15) holds, then But from the backward uniqueness property, and hence Then , the following result holds.; We may then extract a subsequence, again denoted by weakly in , , weakly in 0, ; ( ) .
We deduce from the equality and Combine ( 20) and ( 21) show that there exists a arating closed From the second inequality in (23), must support the set , 0 on 0, .
As the proof of Theorem 3.
such that the optimal control of problem ( 12)-( 17) is bang-bang unique and it is de ined by ( 24), (25) togewhere in this case, the m therefore is p 0 u term ther with (11) (with 0 , 1,2, ,

Scaler Case
Here, we take the case time optimal proble The state is solution of the fo owing equations ll , , , 1, 2 are positive functions in , 2, 2.
The adjoint is solution of the following equations subject to (11) [except in the trivial case where

   
, ; This can be proven in an an as the necessary and sufficient cond , 1,2, ,

Comments
We note that, in this paper, we have chosen to treat a special systems involving Laplace operator just for f the results we described in this paper change on the results to more general for some admissible alogous man er, itions for optim Partial Differenti quations," Springer-Verlag, New York, 1971.http://dx.doi.org/10.1007/978-3-642-65024-6n ality for this problem coincide with (11), ( 16) and (25) (with Most o apply without any parabolic systems involving the following second order operator: ,. ,. ,. ,.

 
In this case, we replace the first eigenvalue of the [12] J. Fleckinger, J. Hernández and F. de Thélin, "On the Existence of Multiple Principal Eigenvalues for Some Indefinite The results chapter 3).problem ( [1] Linear Eigenvalue Problems," Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales.