Time-optimal control problem for co-operative hyperbolic systems involving the Laplace operator

In this paper, the time-optimal control problem for (2 x 2)-co-operative hyperbolic system with variable coefficients and involving Laplace operator is considered. This problem is, steering the initial state ((y(0), y'(0)) so that the observation z(t) hitting a given target set in minimum time. For different cases of the observation, the time-optimal controls are characterized in terms of the adjoint, this characterization (in particular case of the space of controls) is used to derive specific properties of the optimal control (bang-bangness, uniqueness, etc.). These results are extended to the case of (n x n)-co-operative hyperbolic system.