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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 y(0)=u , with control u so that an observation y(t) hitting 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.

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