Optimality conditions for state-constrained Dirichlet boundary control problems
The system is \[ \begin{aligned}{\partial y(t, x) \over \partial t} &= Ay(t, x) + \Phi(t, x, y(t, x)),\quad (t, x) \in (0, T) \times \Omega, \\ y(0, x) &= y_0(x), \qquad x \in \Omega,\end{aligned} \] \((\Omega\) an \(n\)-dimensional domain with boundary \(\Gamma,\) \(A\) a second order elliptic operator). The control \(v(t, x)\) acts on the Dirichlet boundary condition, \[ y(t, x) = v(t, x), (t, x) \in (0, T) \times \Gamma. \] There are pointwise control and state constraints, the latter of the form \[ a(t, x) \leq y(t, x) \leq b(t, x). \] The functional undergoing minimization is \[ J(y, v) = \int_{(0, T) \times \Omega} F(t, x, y(t, x)) dt dx + \int_{(0, T) \times \Gamma} G(t, x, v(t, x)) dt d\sigma. \] As pointed out in this paper, the main difficulty when control is applied on the Dirichlet boundary condition is that the state \(y(t, x)\) is not continuous up to the boundary of \((0, T) \times \Omega,\) which makes it necessary to use multipliers in the dual of the space \(BC((0, T) \times \Omega)\) of bounded continuous functions in \((0, T) \times \Omega,\) or in the dual of \(L^\infty((0, T) \times \Omega).\) The elements of these duals are finitely additive measures. The authors avoid the complexities associated with finitely additive measures by showing that if the functions \(a(t, x)\) and \(b(t, x)\) in the state constraint are continuous up to the boundary, the multipliers can be replaced by bounded Radon measures. Necessary optimality conditions are obtained under Zowe-Kurcyusz type regularity conditions. The optimality conditions are shown to be sufficient if the cost functional is convex and the equation linear.
scientific article; zbMATH DE number 1347769