Joint Coefficient and Solution Estimation for the 1-D Wave Equation: An Observer-Based Solution to Inverse Problems

The estimation of an unknown source coefficient and of the solution to a 1-D wave equation via exponentially convergent state observers is the problem under consideration in this work. The coefficient is assumed to depend on the space variable only and to be polynomial. The main observation information for this inverse problem is the value of the solution to the wave equation in a subinterval of the domain, including also some of its higher order spatial derivatives. In order to estimate the source coefficient, we turn it into a new state as in finite-dimensional parameter identification approaches. However in this infinite-dimensional setting, this requires the introduction of a novel indirect approach involving an infinite-dimensional state transformation. Sufficient conditions allow the design of a composite observer consisting of an internal observer, which estimates in higher regularity spatial norms both the source term and the solution on a subinterval, and a boundary observer, in order to eventually provide the estimation of the solution everywhere. The observer convergence is proven by means of Lyapunov analysis. An extension of this approach to the case of the identification of a potential in the wave equation is finally considered, which is a nonlinear inverse problem since here, in the equation, the unknown coefficient multiplies the solution.