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Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control
(2010)
The paper considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control
problems. By a counter-example of Serrin, it is known that, in the presence of
non-smooth data, a standard weak formulation does not ensure uniqueness for such
equations. Therefore several notions of solution have been developed that guarantee
uniqueness. In this note, we compare different definitions of solutions, namely the ones
of Stampacchia and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution
in the sense of Stampacchia, and prove the existence and uniqueness of solutions in
in case of mixed boundary conditions.
The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.
This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.
We consider an optimal control problem from hyperthermia treatment planning and its barrier regularization. We derive basic results, which lay the groundwork for the computation of optimal solutions via an interior point path-following method. Further, we report on a numerical implementation of such a method and its performance at an example problem.