Abstract
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.
Highlights
The most widely studies of the problems in the mathematical theory of control are the “time optimal” problems
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
Summary
The most widely studies of the problems in the mathematical theory of control are the “time optimal” problems. 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. We will consider a time-optimal control problem for the following n × n co-operative linear hyperbolic system with time delay h and involving. For optimal control problems it is of importance to consider the cases where the control ui belongs to L2 (Q) For these cases, we have the following results: Theorem 1 Let (5), (9) be hold and let yi,0 , yi,1 φi ui be given with yi,0 ∈ H1(Ω), yi,1 ∈ L2(Ω),φi ∈W (I 0),ui ∈ L2(Q). This can be shown by making use of [1] (Remark.1.3 Chapter 4 )
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