EXACT CONTROLLABILITY OF THE DAMPED WAVE-EQUATION

We study the controllability problem for a distributed parameter syste m governed by the damped wave equation u(tt) - 1/rho(x) d/dx (p(x) du/ dx) + 2d(x)u(t) + q(x)u = g(x)f(t), where x is an element of (0, a), w ith the boundary conditions u(0) = 0, (u(x) + hu(t))(a) = 0, h is an e lement of C boolean OR {infinity}. This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping with th e damping coefficient d(x) and with damping (if Re h > 0) or energy pr oduction (if Re h < 0) at one end. (All results extend to the case whe n a similar condition is imposed at the other end as well.) The functi on f(t) is considered as a control. Generalizing well-known results by D. Russell concerning the string with d(x) = 0, we give necessary and sufficient conditions for exact unique controllability and approximat e controllability of the system. Our proofs are based on recent result s by M. Shubov concerning the spectral analysis of a class of nonselfa djoint operators and operator pencils generated by the above equation.