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Optimal control methods in inverse problems and computational processes. (English) Zbl 0986.35123
Let \(W\), \(Y\), \(H_{\text{c}}\), \(H_{\text{ob}}\) be given Banach spaces. The author considers the abstract equation \[ Lu = f_0 + Bv \] where \(f\) is a given data vector; \(v\) is to be determined together with \(u\), \(L\) and \(B\) are closed linear operators, which act from \(W\) into \(Y^*\) and from \(H_{\text{c}}\) into \(Y^*\), respectively. To complete the setting of the problem the equation \[ Cu = g_{\text{ob}} \] is introduced, where the linear operator \(C\) acts from \(W\) into \(H_{\text{ob}}\) and \(g_{\text{ob}}\) is a given element from \(H_{\text{ob}}\). The author considers the following optimal control or variational problem about minimization of the functional \[ J(u(v),v)=\alpha \langle K(v-v_0),v-v_0\rangle+\|Cu(v) - g_{\text{ob}}\|^2 \] with respect to \(v\), where \(\alpha\) is a positive parameter, \( \langle \;,\;\rangle\) is a scalar product, \(K\) is a symmetric positive definite operator, \(v_0\) is a given element of \(H_{\text{c}}\),\(\|\;\|\) is the norm in \(H_{\text{ob}}\). A solvability result for considering variational problems is established, application of this result to the inverse problem of determining the right-hand side of the abstract evolution equation and the boundary function of the transport equation, and to the Stokes problem are given.

MSC:
35R30 Inverse problems for PDEs
49J27 Existence theories for problems in abstract spaces
35Q30 Navier-Stokes equations
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