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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.

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