Convergence of the SQP method for quasilinear parabolic optimal control problems
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Convergence of the SQP method for quasilinear parabolic optimal control problems Fabian Hoppe1 · Ira Neitzel1 Received: 30 October 2019 / Revised: 24 July 2020 / Accepted: 24 July 2020 © The Author(s) 2020
Abstract Based on the theoretical framework recently proposed by Bonifacius and Neitzel (Math Control Relat Fields 8(1):1–34, 2018. https://doi.org/10.3934/mcrf.2018001) we discuss the sequential quadratic programming (SQP) method for the numerical solution of an optimal control problem governed by a quasilinear parabolic partial differential equation. Following well-known techniques, convergence of the method in appropriate function spaces is proven under some common technical restrictions. Particular attention is payed to how the second order sufficient conditions for the optimal control problem and the resulting L 2 -local quadratic growth condition influence the notion of “locality” in the SQP method. Further, a new regularity result for the adjoint state, which is required during the convergence analysis, is proven. Numerical examples illustrate the theoretical results. Keywords Optimal control · Quasilinear parabolic partial differential equation · Sequential quadratic programming · Convergence analysis Mathematics Subject Classification 35K59 · 49K20 · 90C48 · 49N60 · 65K10 · 90C55 · 49M15 · 49M37
1 Overview Optimal control problems governed by linear and semilinear parabolic partial differential equations (PDEs) have been subject to intense research for several years.
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 211504053—SFB 1060.
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Fabian Hoppe [email protected] Ira Neitzel [email protected]
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Institut für Numerische Simulation, Universität Bonn, Endenicher Allee 19b, 53115 Bonn, Germany
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F. Hoppe, I. Neitzel
Existence- and regularity of their solutions is well understood, first order necessary and second order sufficient optimality conditions have been proven, and discretization errors for different types of discretization are available, see e.g. the pioneering work of Lions (1971) concerned with linear PDEs and Hinze et al. (2009), or Tröltzsch (2010) for a recent overview covering theoretical and numerical aspects of both linear and nonlinear problems. Recently, optimal control of quasilinear parabolic equations was addressed by Bonifacius and Neitzel (2018), Casas and Chrysafinos (2018), and Meinlschmidt et al. (2017a, b), Meinlschmidt and Rehberg (2016). The functional analytic framework for the analysis of the state equation is provided by the concept of maximal parabolic regularity of nonautonomous operators, see e.g. the work of Amann (2004, 2003, 2005), Meinlschmidt and Rehberg (2016), Haller-Dintelmann and Rehberg (2009), or further references in Bonifacius and Neitzel (2018). The highly non-trivial existence and regularity theory for solutions of the underlying PDE poses the main difficulty in the theoretical analysis of such problems. For a discussion of previous literature concerning optimal control of quasilinear PDEs
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