Exact Controllability of a Damped Wave Equation with Distributed Controls

We study the controllability problem of the one-dimensional damped wave equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\rho {\text{(}}x{\text{)}}u_{tt} - \frac{d}{{dx}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \left( {p(x)u_x } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2d(x)\rho (x)u_t + q(x)\rho (x)u = h(x,t),{\text{ }}x \in {\text{(0,1)}}$$ \end{document} 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.