Minimum Time Problem for n×n Co-operative Hyperbolic Lag Systems

In this paper, a minimum time problem for n×n co-operative hyperbolic systems involving Laplace operator and with time-delay is considered. First, the existence of a unique solution of such hyperbolic system with time-delay is proved. Then necessary conditions of a minimum time control are derived in the form of maximum principle. Finally the bang-bang principle and the approximate controllability conditions are investigated.


Introduction
The most widely studies of the problems in the mathematical theory of control are the "time optimal" problems. The simple version is that steering the initial state 0 y in a Hilbert space H to hitting a target set H K ⊂ in minimum time , with control subject to constraints ( H U u ⊂ ∈ ). In this paper we will focus our attention on some special aspects of minimum time problems for co-operative hyperbolic systems with time delay. In order to explain the results we have in mind, it is convenient to consider the abstract form: Let V and H be two real Hilbert spaces such that V is a dense subspace of H .Identifying the dual of H with H we may consider In practical applications, the behavior of many dynamical systems which describes a state of time-optimal control problems depends upon their past histories. This phenomenon can be induced by the presence of time delays. Due to the inherent difficulties in solving control problems with time delays, the progress in this area has been slow. Here, we mention the work of Wang [4], where the time optimal control for a class of ordinary differential-difference equation with time lag was considered. Also, we mention the work of Knowles [5], where a Time optimal control of parabolic systems with boundary condition involving time delays was considered and it is shown that the optimal control is characterized in terms of an adjoint system and it is of the bang-bang type.
Time-optimal control of distributed parameter systems governed by a system of hyperbolic equations is of special importance for the active control of structural systems for which the equations of motion are generally expressed by hyperbolic differential equations. A typical application of a hyperbolic equation is the vibrating system.Time-optimal control of distributed parameter systems governed by a system of hyperbolic equations have been studied in many papers, we mention only [6], [7] in which time optimal distributed control problems of vibrating systems has been studied. In our papers [8][9][10][11], the results in [6] and [7] have been extended to the time optimal control problems for systems governed by n n × hyperbolic systems, involving laplace operator with different cases of observations.
In this paper, we will consider a time-optimal control problem for the following n n × co-operative linear hyperbolic system with time delay h and involving Laplace operator (here and everywhere below the vectors are denoted by bold letters.): This problem is, steering the initial vector state (0) First, we establish the well posedness of the system (4) under conditions on the coefficients stated by the principal eigenvalue of the Laplace eigenvalue problem. Then, we formulate a time optimal control problem and we derive the necessary and sufficient conditions which the optimal control must satisfy in terms of the adjoint.
For simplicity, We introduce the following notations: for For optimal control problems it is of importance to consider the cases where the control i u belongs to ). ( 2 Q L For these cases, we have the following results: Theorem 1 Let (5), (9) be hold and let

Co-operative Hyperbolic Systems
We will denote by ) ; ( u y t to the unique solution of (4), at time t corresponding to a given control n U ε ∈ u and a given functions In this section, we consider the following first time-optimal control problem with control v acts in velocity initial condition and position observation : In order for the problem (TOP) to be well posed, we assume the following (11) The following result holds . Theorem 2 If (5), (9) (11) ). Moreover Combine (14) and (15) The first term in the right hand side of (22)can be rewrite as We now summarize the foregoing result Theorem 3 We assume that (5), (9) hold. Then there exist the adjoint state

Bang-Bang Control and Controllability
The maximum conditions (24) of the optimal control leads to the following result: Theorem 4 (Bang-Bang theorem) Let the hypothesists of Theorem.3 be satisfied. Then the optimal control of (TOP) is unique and bang -bang, that is Proof. The proof will follow from Theorem3 if we can show that We shall show this fact by contradiction. Therefore, we suppose that Note that, in the cylinder We have just shown that, then by analyticity and continuity as before, Substituting (26) in to (25) gives consequently, by backward uniqueness [12], We can repeat this argument until 0, = ) ; ( 0 0 u τ i p which leads to a contradiction. Since n U ε is strictly convex, then the optimal control is unique. This is complete the proof. With regard to controllability assumption (10), We can show quit easily that (4)

Scalar Cases
In this section, we give some special cases. Case I: Coupled system with time delay Here, we take the case where 2, = n in this case, the time optimal problem therefore is

Comments
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 : ,.) In this case, we replace the first eigenvalue of the Laplace operator by the first eigenvalue of the operator L (see [9]).
• In this paper, we have chosen to treat a co-operative hyperbolic systems with Dirichlet boundary onditions.The results can be extended to the case of n n × co-operative Also, we can take another observation (see [7]).
• The results in this paper, carry over to the optimal control problems with fixed -time ( [1] chapter 4 ), for example, he results of (TOP) carry over to the fixed -time problem ).
• As a final coment, we note that the control problem for the second order evolution system (4) can be reduced to a similar control problem first order system; in the usual form. However, the existing results on the time-optimal problem ([1], [10], [11]) pertain to the case where the observation is only one case (position-velocity ) but here we can take different cases.