# zbMATH — the first resource for mathematics

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
Full Text: