Necessary conditions for optimality in the identification of elliptic systems with parameter constraints. (English) Zbl 0851.93025
Summary: We consider the problems of identifying the parameters \(a_{ij} (x)\), \(b_i (x)\), \(c(x)\) in a 2nd order, linear, uniformly elliptic equation,
\[
-\partial_i \bigl( a_{ij} (x) \partial_j u \bigl) + b_i(x) \partial_i u + c(x) u = f(x), \text{ in } \Omega, \qquad \partial_\nu u |_{\partial \Omega} = \varphi (s), \quad s \in \partial \Omega,
\]
on the basis of measurement data
\[
u(s) = z(s), \quad s \in B \subset \partial \Omega,
\]
with an equality constraint and inequality constraints on the parameters. The cost functionals are one-sided Gâteaux differentiable with respect to the state variables and the parameters. Using the Duboviskij-Milyutin lemma, we get maximum principles for the identification problems, which are necessary conditions for the existence of optimal parameters.
MSC:
| 93B30 | System identification |
| 93C20 | Control/observation systems governed by partial differential equations |
| 35R30 | Inverse problems for PDEs |
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