##
**Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations.**
*(English)*
Zbl 0893.49017

The state equation is
\[
{\partial \over \partial t} y(t, x) + Ay(t, x) + a_0(t, x, y(t, x)) = 0
\]
where \(A\) is a linear elliptic differential operator in a domain \(\Omega \subset{\mathbb{R}}^m\) with boundary \(\Gamma.\) The (controlled) boundary condition is
\[
\partial_{\nu_A} y(t, x) = f(t, x, y(t, x), u(t, x))
\]
where \(\partial_{\nu_A}\) is a directional derivative on \(\Gamma\) associated with \(A.\) The problem includes constraints on the control \(u(t, x)\) as well as on the state \(y(t, x),\) and the cost functional to be minimized in a time interval \(0 \leq t \leq T\) is
\[
J(u) = \int_{(0, T) \times \Omega} L(t, x, y_u(t, x)) dt dx + \int_{(0, T) \times \Sigma} l(t, x, y_u(t, x), u(t, x)) dt d\sigma(x).
\]
The author derives a maximum principle of Pontryagin’s type for controls minimizing the cost functional under all constraints. Besides the new result, the paper is a very good survey of the different approaches to this sort of problems (for instance, spike vs. patch or diffuse perturbations) and of the difficulties associated with the adjoint variational equation, whose inhomogeneous term is a measure rather than a function.

Reviewer: H.O.Fattorini

### MSC:

49K20 | Optimality conditions for problems involving partial differential equations |

35J20 | Variational methods for second-order elliptic equations |

93C20 | Control/observation systems governed by partial differential equations |