×

On a nonclassical perturbed boundary optimal control system. (English) Zbl 0871.49007

For the nonclassical perturbed boundary optimal control problem (which arises in aerodynamics): \[ -\Delta y_\varepsilon(v)= 0\quad\text{in } \Omega, \]
\[ {\partial \over \partial \nu}y_\varepsilon(v)+ \varepsilon y_\varepsilon(v) =v\quad\text{on } \Gamma, \]
\[ J_\varepsilon (u_\varepsilon)= \min_{v\in U_{ad}} |y_\varepsilon (v)- z|^2_{L^2 (\Gamma)}, \] \(y_\varepsilon(v) \in H^1(\Omega)\), \(U_{ad}\) is a closed subspace of \(L^2 (\Gamma)\) where \(\Omega\subset R^n\) is a regular bounded open set with the smooth boundary \(\Gamma\) and \(z\) is a given element of \(L^2(\Gamma)\), the following assertions are proved:
i) If \(U_{ad}\) has finite dimension then there exists the unique optimal control \(u_\varepsilon\) and \((u_\varepsilon, y_\varepsilon (u_\varepsilon))\) is convergent with \(\varepsilon \to 0\) in \(L^2(\Gamma) \times H^1(\Omega)\),
ii) If \(U_{ad}\) has infinite dimension then the solution to the above optimal control problem does not exist (an example is given).

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B37 PDE in connection with control problems (MSC2000)
93C20 Control/observation systems governed by partial differential equations
PDF BibTeX XML Cite