Exact boundary and distributed controllability of radial damped wave equation

Abstract

We study the zero, closed loop and periodic controllability problems for the distributed parameter systems governed by the radial wave equations containing damping terms both in the equation and in the boundary conditions. These equations are obtained by the separation of variables in the spherical coordinates from the 3-dimensional damped equation with spacially nonhomogeneous spherically symmetric coefficients. We consider two types of controls: a) the distributed controls implemented as forcing terms in the right-hand sides of the equations and b) the boundary controls implemented through the boundary conditions. Applying the spectral decomposition method, we give the necessary and sufficient conditions for the exact controllability of the systems and provide explicit formulas for the controls for all three aforementioned problems and for both types of controls. The proofs are based on our recent results concerning the spectral analysis for the class of nonselfadjoint operators and operator pencils generated by the above equations and the boundary conditions. These operators are the dynamic generators of the systems in the energy spaces of two-component initial data. We do not restrict our analysis to the case when the spectra of the dynamic generators are simple and assume that they may have associated vectors, i.e., the algebraic multiplicities of their eigenvalues are greater than one.