Estimation of time dependent parameters in general parabolic evolution systems. (English) Zbl 0869.35047

Ackleh, A. S.; Fitzpatrick, B. G.

J. Math. Anal. Appl. 203, No. 2, 464-480 (1996).

The authors present a unified theoretical and computational approach for estimating time dependent parameters in abstract parabolic systems. Results are proved initially, within the framework of a Gelfand triple \(V\hookrightarrow H\hookrightarrow V^*\), for the abstract non-autonomous equation \[ \dot u(t,q)= A(t,q)u(t,q)+ F(t,u(t,q),q),\quad u(0,q)= u_0(q).\tag{1} \] In (1) it is assumed that the parameter \(q\) belongs to a compact separable metric space, the linear operator \(A(t,q)\) is determined by a (suitably restricted) time and parameter dependent sesquilinear form on \(V\), and the nonlinear term \(F\) satisfies certain continuity conditions. The existence and uniqueness of a solution to a weak formulation of (1) is established and a convergence theory for least squares based parameter estimation is produced. The general theory is then applied to the specific cases of a contaminated groundwater model and the Euler-Bernoulli beam equation.

Reviewer: W.Lamb (Glasgow)

MSC:

35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
34G20	Nonlinear differential equations in abstract spaces
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Keywords:

abstract parabolic systems; contaminated groundwater model; Euler-Bernoulli beam equation