Nonselfadjoint operators and controllability of damped string

Abstract

We study the controllability problem for a distributed parameter system governed by the damped wave equation u/sub tt/$ -/sub /spl rho/(x/)//sup 1//sub dx///sup d/(p(x)/sub dx///sup du/)+2d(x)u/sub t/+q(x)u=g(x)f(t), where x/spl isin/(0,a), with the boundary conditions (u/sub x/+ku/sub t/)(0,t)=0,(u/sub x/+hu/sub t/)(a,t)=0, h,k/spl isin/C/spl cup/{/spl infin/}. 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) 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.