A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints
- PDF / 1,094,851 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 93 Downloads / 177 Views
A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints Samuel Amstutz · Antoine Laurain
Received: 10 December 2012 / Published online: 29 March 2013 © Springer Science+Business Media New York 2013
Abstract In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0 ≤ u ≤ 1 and u = m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0–1 values, which amounts to solving a topology optimization problem with volume constraint. Keywords Optimal control · Topology optimization · Semilinear equation · Semismooth Newton method · Volume constraint
1 Introduction This paper is dedicated to the numerical solution of minimization problems of the form min
(u,y)∈Uad ×Y
J (y)
subject to E(u, y) = 0,
S. Amstutz () Laboratoire de Mathématiques d’Avignon, Faculté des Sciences, Université d’Avignon, 33 rue Louis Pasteur, 84000 Avignon, France e-mail: [email protected] A. Laurain Department of Mathematics, Technical University of Berlin, Berlin, Germany e-mail: [email protected]
(1)
370
S. Amstutz, A. Laurain
where J : Y → R and E : U × Y → Z are appropriate functionals, Y and Z are Banach spaces, the sets U and Uad are defined by U := u ∈ L2 (D), 0 ≤ u ≤ 1 a.e. in D , Uad := u ∈ U, u = m , 0 < m < |D|, D
and D is a bounded domain of RN , N ∈ {2, 3}, with N -dimensional Lebesgue measure |D|. In [2] a semismooth Newton method was introduced for a control problem subject to a linear elliptic state equation and an L1 control cost, with the feature that the control u, a priori searched for within U , eventually takes 0–1 values. Such a problem is actually a topology optimization problem [1, 4] since u may be written as the characteristic function of a measurable domain Ω ⊂ D. We speak of topology optimization rather than shape optimization since the topology of Ω is not imposed and may be complex. The control cost D u is interpreted as a volume penalization, which is standard in topology optimization. In the present paper we extend the approach of [2] mainly in two directions. Firstly, the volume term is now treated as an equality constraint instead of a simple penalization. Secondly, we consider a class of semilinear state equations, for which the optimal controls are not necessarily in 0–1. Nonsmooth control costs or constraints such as the L1 -norm usually lead to optimal controls whose structure is fundamentally different than when using smooth control costs such as Lp norms wit
Data Loading...
