Determining the Acoustic Impedance in the 1-D Wave Equation via an Optimal Control Problem

Abstract

This paper deals with a coefficient optimal control problem for the one-dimensional (1-D) wave equation with nonhomogeneous boundary periodic inputs. A main concept is the notion of "weak solution" for the 1-D wave equation with T-periodic conditions (Definition 2.1). For T=(2k+1)/ p the weak solution (hyperbolic) operator A(u) (see 2.20) has important properties such as closed range R(A(u)), nontrival null space N(A(u)) (resonant case)---see Propositions 2.1 and 2.2. However, the u-dependence of R(A(u)) and N(A(u)) gives rise to major difficulties. The maximum principle (Theorem 4.1) can be viewed as information (necessary conditions) on the optimal acoustic impedance function u* (theimpedance for which the corresponding seismic waves have a minimal effect given as a cost functional).