IDENTIFICATION FOR PARABOLIC DISTRIBUTED-PARAMETER SYSTEMS WITH CONSTRAINTS ON THE PARAMETERS AND THE STATE

We consider the problems for identifying the parameters a(11)(x, t),.. .,a(mm)(x, t) and c(x, t) involved in a second-order, linear, uniforml y parabolic equation partial derivative(t)u - partial derivative(i)(a( ij) (x, t)partial derivative(j)u) + bi(x, t)partial derivative(i)u + c (x, t)u = f(x, t) in Omega x (0, T), u\(partial derivative Omega) = g, u\(t = 0) = u(0)(x), x is an element of Omega. on the basis of noisy measurement data z(x) = u(x, T) + w(x), x is an element of Omega with equality and inequality constraints on the parameters and the state va riable. The cost functionals are (one-sided) Gateaux-differentiable wi th respect to the state variables and the parameters. Using the Dubovi skii-Miljutin lemma we get the two maximum principles for the two iden tification problems, respectively, i.e., the necessary conditions for the existence of optimal parameters.