Structural identification of an unknown source term in a heat equation.

*(English)*Zbl 0917.35156The authors are concerned with the structural identification of a state-dependent source term in a nonlinear diffusion equation. It means that \(F(u)\) is a function of the state variable only and does not depend explicitly on position or time. In the context of heat conduction or diffusion the unknown function \(F(u)\) is interpreted as a heat or material source while in a chemical or biochemical application \(F\) may be interpreted as a reaction term. In a control theory setting \(F\) represents a nonlinear feedback law.

The paper is organized as follows. Section 1 examines the initial boundary value problem that models a physical experiment, in which a one-dimensional heat equation containing a state-dependent source term is driven by controlling the fluxes (i.e. the first-order spatial derivatives) on the boundaries. Several properties of the solution as well as a basic integral identity are derived in this section. In section 2 the inverse problem is formulated and the identity is used to show that the measured output uniquely determines the unknown source term in an appropriate class of polygonal source functions. The section also contains the proof that a source term that minimizes an output least squares functional that is based on the measured output must also be a solution of the associated inverse problem. Section 3 describes an algorithm for the explicit construction of an appropriate inverse for the source to data mapping. In section 4 the results of some numerical experiments are discussed.

The paper is organized as follows. Section 1 examines the initial boundary value problem that models a physical experiment, in which a one-dimensional heat equation containing a state-dependent source term is driven by controlling the fluxes (i.e. the first-order spatial derivatives) on the boundaries. Several properties of the solution as well as a basic integral identity are derived in this section. In section 2 the inverse problem is formulated and the identity is used to show that the measured output uniquely determines the unknown source term in an appropriate class of polygonal source functions. The section also contains the proof that a source term that minimizes an output least squares functional that is based on the measured output must also be a solution of the associated inverse problem. Section 3 describes an algorithm for the explicit construction of an appropriate inverse for the source to data mapping. In section 4 the results of some numerical experiments are discussed.

Reviewer: Gabriel Dimitriu (Iaşi)