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).


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