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Shape optimization of control problems described by wave equations

Nowakowski A.

Control and Cybernetics

2008

Vol. 37, no 4

1045-1055

The control problem with multidimensional integral functional under wave type constraints for control is considered. Next a type of deformation with control of the domain is described and then we define suitable shape functional. Having denned trajectory and control of deformation dual dynamic programming tools are applied to derive optimality condition for the shape functional with respect to that deformation.

Optimality conditions for state-constrained PDE control problems with time-dependent controls

Los Reyes J. C. de, Merino P., Rehberg F., Troltzsch F.

Control and Cybernetics

2008

Vol. 37, no 1

5-38

The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rm -vector-valued functions of time, while they are vectors of Rm in elliptic problems. Under natural assumptions, first- and second-order sufficient optimality conditions are derived. The main result is the extension of second-order sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimization. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.

Second derivatives and sufficient optimality conditions for shape funetionals

Eppler K.

Control and Cybernetics

2000

Vol. 29, no 2

485-511

For sorne heuristic approaches to boundary variation in shape optimization the computation of second derivatives of domain and boundary integral functionals, their symmetry and a comparison to the velocity field or material derivative method are discussed. Moreover, for these approaches the functionals are Frechet-differentiable in some sense, because at least a local embedding into a Banach space problem is possible. This allows the discussion of sufficient condition in terms of a coercivity assumption on the second Frechet-derivative. The theory is illustrated by a discussion of the famous Dido problem.