## 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