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 to hit a target set in minimum time, with control subject to constraints.

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 be two real Hilbert spaces such that is a dense subspace of Identifying the dual of with we may consider 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 and

Then, from [1] and [2], for and being a bounded linear operator on the following abstract systems:

have a unique weak solution such that We shall denote by the unique solution of the Equation (2) corresponding to the control u. The time optimal control problem that we shall concern reads:

where is a given subset of which is called the target set of the Problem (3). A control is called a time optimal control if and if there is a number such that and

We call the number as the optimal time for the time optimal control Problem (3).

Three questions (problems) arise naturally in connection with this problem:

1) Is there a control and such that? (this is an approximate controllability problem).

2) Assume that the answer to 1) is in the affirmative and

Is there a control which steers to hit a target set in minimum time?

3) If exists, is it unique? What additional properties does it have?

Let be a bounded open domain with smooth boundary and set 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 is a given function in and is a closed bounded set in 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 and certain controlled systems. These results will be stated as follows. Consider the following controlled system

where is a real number. Let be the eigenvalues of with the Dirichlet boundary condition and be the corresponding eigenfunctions, which forms an orthogonal basis of We take the target set to be the origin in and the control set to be the set

where is a positive number, namely, the closed ball in centered at 0 and of radius It was proved that if and then the corresponding time optimal control problem has at least one solution if and only if

More early, in the works [5-7], 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 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 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.

Let be the usual Sobolev space of order one which consists of all whose distributional derivatives and with the scalar product norm

We have the following dense embedding chain [11]

where is the dual of

Here and everywhere below the vectors are denoted by bold letters. For and, let us define a family of continues bilinear forms

where

The bilinear form (7) can be but in the operator form:

where is matrix operator which maps onto and takes the form

.

Lemma 2.1. 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 [12]. ,.

Now, let

Lemma 2.2. If (8) and (10) hold then, the bilinear form (7) satisfy the Gårding inequality

Proof. In fact

By Cauchy Schwarz inequality and (9), we obtain

From (10) we have

Add 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:

Theorem 2.3. If (8) and (10) hold, then there exist a unique solution

satisfying the following system:

Moreover is continuous from

We denote the unique solution of (11), at time for each control by Occasionally, we write when the explicit dependence on is required. We can now formulate the time optimal control problem corresponding to the cooperative parabolic system (11):

with constraints

and and are given.

Theorem 3.1. If (8) and (10) are hold, then the system whose state is given by (11) is controllable, i.e.,

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 in any finite time 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 by the solution of the following system

where is the adjoint of which is defined by

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 and integrate by parts from 0 to we obtain the following identity:

and so, if (15) holds, then

hence But from the backward uniqueness property, and hence ,.

Now set

Then , the following result holds.

Theorem 3.2. If (8) and (10) are hold, then there exist an admissible control to the problem (12)-(17), which steering to hitting a target set in minimum time (defined by (17)). Moreover

Proof. Fixe we can choose and admissible controls such that

Set Since is bounded, we may verify that ranges in a bounded set in

.

We may then extract a subsequence, again denoted by such that

We deduce from the equality

that

and

But

Now from (19)

and

Combine (20) and (21) show that

and so, as is closed and convex, hence weakly closed. This shows that is reached in time by admissible control.

For the second part of the theorem, really, from Theorem 2.3, the mapping from is continuous for each fixed and so for any by minimality of

Using Theorem 2.3, it is easy to verify that the mapping, from, is continuous and linear. then, the set

is the image under a linear mapping of a convex set hence is convex. Thus we have

and (boundary of) . Since (from (14)) so there exists a closed hyperplane separating and containing, i.e. there is a nonzero such as

From the second inequality in (23), must support the set at i.e.

and since is a Hilbert space, must be of the form

Dividing the inequality (23) by gives the desired result. ,.

Now Inequality (18) can be interpreted as follows: let us introduce the adjoint state by the solution of the following system

As the proof of Theorem 3.1, we multiply the first equation int (24) by and integrate by parts from to we obtain the following identity:

hence condition (18) becomes

Using controllability condition (14), the backward uniqueness property implies then the optimal control is bang-bang, i.e., and since

is strictly convex, then the optimal control is unique. We have thus proved:

Theorem 3.3. If (8) and (10) are hold, then there exist the adjoint state

such that the optimal control of problem (12)-(17) is bang-bang unique and it is determined by (24), (25) together with (11) (with).

Here, we take the case where in this case, the time optimal problem therefore is

The state is solution of the following equations

with

The adjoint is solution of the following equations

The maximum condition is

We note that, in this paper, we have chosen to treat a special systems involving Laplace operator just for simplicity. 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 LegendreHadamard ellipticity condition

for all and some constants

In this case, we replace the first eigenvalue of the Laplace operator by the first eigenvalue of the operator (see [12]).

In this paper, we have chosen to treat a co-operative parabolic system with Dirichlet boundary conditions. The results can be extended to the case of cooperative parabolic system with Neumann boundary conditions: if we take instead of we have to replace the Dirichlet boundary conditions on the boundary by Neumann boundary conditions where is the outward normal.

The results in this paper carry over to the fixed-time problem ([1] chapter 3).

subject to (11) [except in the trivial case where for some admissible control]. This can be proven in an analogous manner, as the necessary and sufficient conditions for optimality for this problem coincide with (11), (16) and (25) (with).