Spectrum rearrangement via feedback control for nonhomogeneous damped string

The solution of the pole reassignment control problem for a distributed parameter system is presented. The system is a vibrating string with spatially non-homogeneous density and modulus of elasticity coefficients and with both distributed and boundary damping. The string equation and the boundary conditions are reduced to a single linear evolution equation in the Hilbert space of two-component Cauchy data equipped with energy metric. It is the dynamics generator of this equation, whose spectrum is rearranged with a feedback control. The solution is based on the fact that the dynamics generator, its finite-rank perturbations and the adjoints to these operators are Riesz spectral, i.e. their sets of the generalized eigenvectors form (non-orthogonal) Riesz bases in the state space. The approach extends to any system, whose dynamics generator has similar properties.