Sensitivity analysis of hyperbolic optimal control problems

Abstract

The aim of this paper is to perform sensitivity analysis of optimal control problems defined for the wave equation. The small parameter describes the size of an imperfection in the form of a small hole or cavity in the geometrical domain of integration. The initial state equation in the singularly perturbed domain is replaced by the equation in a smooth domain. The imperfection is replaced by its approximation defined by a suitable Steklov's type differential operator. For approximate optimal control problems the well-posedness is shown. One term asymptotics of optimal control are derived and justified for the approximate model. The key role in the arguments is played by the so called "hidden regularity" of boundary traces generated by hyperbolic solutions.

Application of topological derivative to accelerate genetic algorithm in shape optimization of coupled models

“…Thus, according to Sokołowski andŻochowski (2005) and Kowalewski et al (2012) we have the following expansion of the Steklov-Poincaré operator:…”

On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions

“…There has been much attention to studies related to optimal control problems involving hyperbolic problems [1]. There have been many studies about optimal control for hyperbolic systems which are considered [2][3][4].…”

Topology optimization of quasistatic contact problems

“…In recent years the topological derivative method (Garreau et al, 2001;Kowalewski et al, 2010;Nazarov and Sokołowski, 2003;Novotny et al, 2005;Sokołowski andŻochowski, 1999; has emerged as an attractive alternative to analyze and solve numerically topology optimization problems, especially of elastic structures, without employing the homogenization approach. The topological derivative gives an indication on the sensitivity of the shape functional with respect to the nucleation of a small hole or a cavity, or more generally a small defect in a geometrical domain Ω around a given point.…”

“…A few papers only (e.g., Amstuz et al, 2008;Kowalewski et al, 2010) address this issue for the shape functionals depending on a solution to time-dependent boundary value problems. One of the reasons is that the approaches useful for stationary boundary value problems fail for evolution problems (Kowalewski et al, 2010). The approach of Amstuz et al (2008) extends the ideas of Garreau et al (2001).…”

“…A polarization matrix is used to calculate the topological derivatives of the different shape functionals depending on solutions to heat or wave equations. The paper by Kowalewski et al (2010) deals with the sensitivity analysis of the optimal control problem for the wave equation with respect to the small hole or cavity in the geometrical domain using the "hidden regularity" argument for boundary traces as well as the expansion of the elliptic Steklov-Poincaré operator. Evolution boundary value problems may be also considered as the eigenvalue problem.…”