Stability Estimates in State-Estimation for a Heat Process

Abstract

This paper deals with a heat equation with the homogeneous Dirichlet boundary condition:

$$\left\{ {\begin{array}{*{20}{c}} {{u_t}(t,x) = \Delta u(t,x),0 < t < T,X \in \Omega,} \\ {u(t,x) = 0,0 < t < T,x \in \partial \Omega,}\end{array}} \right.$$

where $\Omega \subset {R^n}$ is a bounded domain with smooth boundary $\partial \Omega$. For given θ ∈ [0, T), we discuss a problem of determining u(θ, •) from $u{{|}_{{(0,T) \times \omega }}},$ where $\omega \subset \Omega$ is an arbitrary subdomain. Our main result is the Lipschitz stability estimate in the case of 0 < θ < T and conditional stability of logarithmic rate in the case of θ = 0. The proof relies on a Carleman estimate for a parabolic equation.

Jiangxi Provincial Natural Scientific Foundation and National Natural Scientific Foundation of P. R. China

Sanwa Systems Development Co. Ltd(Tokyo, Japan).