Nonselfadjoint operators and controllability of damped string

We study the controllability problem for a distributed parameter system governed by the damped wave equation u_tt$ -_ρ(x/)¹_dx/^d(p(x)_dx /^du)+2d(x)u_t+q(x)u=g(x)f(t), where x∈(0,a), with the boundary conditions (u_x+ku_t)(0,t)=0,(u_x+hu_t )(a,t)=0, h,k∈C∪{∞}. This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping with the damping coefficient d(x) and with the damping (if Re k<0 and Re h>0) or energy production (if Re k>0 and Re h<0) through the boundary. The function f(t) is considered as a control. Generalizing well known results by Russell (1967) concerning the string with d(x)=0, we give necessary and sufficient conditions for exact and approximate controllability of the system. Our proofs are based on recent results by Shubov concerning the spectral analysis of a class of nonselfadjoint operators and operator pencils generated by the above equation