Exact Controllability of a Damped Wave Equation with Distributed Controls

We study the controllability problem of the one-dimensional damped wave equation $$rho {text{(}}x{text{)}}u_{tt} - frac{d}{{dx}}{kern 1pt} {kern 1pt} {kern 1pt} left( {p(x)u_x } right){kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} + {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} {kern 1pt} 2d(x)rho (x)u_t + q(x)rho (x)u = h(x,t),{text{ }}x in {text{(0,1)}}$$ This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping. It is proved that the solution can be exactly controlled in finite time by means of distributed control forces h which vanish outside of any fixed nonempty subinterval of (0, 1). Moreover the optimal time of controllability is given.