Logarithmic stability in determining two coefficients in a dissipative wave equation. Extensions to clamped Euler–Bernoulli beam and heat equations

Abstract

We are concerned with the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. We establish stability estimates of logarithmic type when the measurements are given by the operator who maps the initial condition to Neumann boundary trace of the solution of the corresponding initial–boundary value problem. We build a method combining an observability inequality together with a spectral decomposition. We also apply this method to a clamped Euler–Bernoulli beam equation. Finally, we indicate how the present approach can be adapted to a heat equation.