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**Shape sensitivity analysis of boundary optimal control problems for parabolic systems.**
*(English)*
Zbl 0663.49012

A family of parabolic equations defined on smooth domains depending on a parameter is considered. The domains are described by a vector field. For each fixed value of the parameter the parabolic boundary optimal control problem is considered. The cost functional is quadratic. The control is executed either through Neumann or Dirichlet boundary conditions.

The differential properties of the optimal control treated as a function of the parameter are investigated. The so called Euler and Lagrange derivatives of the control in the direction of the vector field are characterized as the solutions of some auxiliary optimal control problems.

In the proof the material derivative method is used combined with some results on directional differentiability of the projection onto a closed convex set in a Hilbert space.

The differential properties of the optimal control treated as a function of the parameter are investigated. The so called Euler and Lagrange derivatives of the control in the direction of the vector field are characterized as the solutions of some auxiliary optimal control problems.

In the proof the material derivative method is used combined with some results on directional differentiability of the projection onto a closed convex set in a Hilbert space.

Reviewer: K.Malanowski

### MSC:

49K40 | Sensitivity, stability, well-posedness |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49J52 | Nonsmooth analysis |