Inverse Problems for Damped Wave Equations

Inverse problems for wave equations have been extensively studied in the last 50 years due to a large number of engineering and technological applications. The objective of this chapter is to present an analysis of the two basic inverse coefficient problems related to 1D damped wave equations m(x)utt + μ(x)ut = (r(x)ux)x and \(m(x) u_{tt}+\mu (x)u_{t}=\left (r(x)u_{x}\right )_{x}+q(x)u\), \(\Omega _{T}:=\{(x,t)\in \mathbb {R}^2\,:\,x\in (0,\ell ),\, t\in (0,T)\}\), when l > 0 and T > 0 are finite. Here and below μ(x) ≥ 0 is the damping coefficient. Specifically, we study the inverse problem of identifying the unknown principal coefficient r(x) from Dirichlet-to-Neumann operator, and the inverse problem of recovering the unknown potential q(x) from either from Neumann-to-Dirichlet or Dirichlet-to-Neumann operators.