Energy decay rates and equilibrium state properties for a distributed parameter identification algorithm

The author considers an inverse problem, i.e., the identification of the position-dependent leading coefficient appearing in elliptic ordinary differential equations (a scalar) and partial differential equations (a diagonal matrix). He examines the identification algorithm based on the substitution or comparison model method. As suggested by the method of stationarization, he models the algorithm by a nonlinear dynamical system, the state of which is the unknown coefficient. With reference to some inverse problems in one and two spatial dimensions in the continuum case, the state equations are obtained. The author determines the system's energy function, its decay rate, and the influence constraints have on it. He then considers the admissible equilibrium states. If an admissible coefficient, i.e., a solution to the inverse problem, exists, it is generally not unique: the (admissible) equilibrium state depends on the initial state and on a control term, some preliminary properties of which the author specifies in some special cases.<>