A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems
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A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems Veronika Karl1 · Ira Neitzel2 · Daniel Wachsmuth1 Received: 20 June 2018 © The Author(s) 2020
Abstract In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented. Keywords Optimal control · Semilinear elliptic operators · State constraints · Augmented Lagrange method Mathematics Subject Classification 49M20 · 65K10 · 90C30
This research was supported by the German Research Foundation (DFG) within the priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) under Grant Number WA 3626/3-1 and NE 1941/1-1. * Veronika Karl [email protected]‑wuerzburg.de Ira Neitzel [email protected]‑bonn.de Daniel Wachsmuth [email protected]‑wuerzburg.de 1
Institut für Mathematik, Universität Würzburg, Emil‑Fischer‑Str. 30, 97074 Würzburg, Germany
2
Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany
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1 Introduction In this paper, the solution of an optimal control problem subject to a semilinear elliptic state equation and pointwise control and state constraints will be studied. The control problem is non-convex due to the nonlinearity of the state equation. The problem under consideration is given by
min J(y, u) ∶=
𝛼 1 ||y − yd ||2L2 (Ω) + ||u||2L2 (Ω) 2 2
(1)
subject to
Ay + d(y) 𝜕𝜈A y y u
=u in Ω, =0 on Γ, ≤𝜓 in Ω, ∈ Uad .
(2)
Here, A denotes a second-order elliptic operator, while d(y) is a nonlinear term in y. The setting of the optimal control problem will be made precise in Sect. 2. Optimal control problems with pointwise state constraints suffer from low regularity of the respective Lagrange multipliers, see [4, 6] for Dirichlet problems and [5] for Neumann problems. The multiplier 𝜇̄ associated to the state constraint is a Borel measure. Under additional assumptions it has been proven in [8] that the multiplier satisfies H −1 (Ω)-regularity. These assumptions are satisfied, e.g., for 𝜓 constant. In contrast to non-convex problems, convex state constrained optimal control problems allow a much simpler convergence analysis of optimization algorithms and have therefore been studied extensively during the last years. To give just a brief insight into the literature of convex problems, we want to mention the common approaches given by penalization based methods [13–16, 19] possibly with fixed shift, interior point
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