Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Richard Krug; Günter Leugering; Alexander Martin; Martin Schmidt; Dieter Weninger

doi:10.2478/candc-2021-0026

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Time-domain decomposition for optimal control problems governed by semilinear hyperbolic systems with mixed two-point boundary conditions

Control and Cybernetics

Volume 50 (2021): Issue 4 (December 2021)

By: Richard Krug, Günter Leugering, Alexander Martin, Martin Schmidt and Dieter Weninger

Open Access

|Jun 2022

Abstract

In this article, we study the time-domain decomposition of optimal control problems for systems of semilinear hyperbolic equations and provide an in-depth well-posedness analysis. This is a continuation of our work, Krug et al. (2021) in that we now consider mixed two-point boundary value problems. The more general boundary conditions significantly enlarge the scope of applications, e.g., to hyperbolic problems on metric graphs with cycles. We design an iterative method based on the optimality systems that can be interpreted as a decomposition method for the original optimal control problem into virtual control problems on smaller time domains.

References

Barker, A. T. and Stoll, M. (2015) Domain decomposition in time for PDE-constrained optimization. In: Computer Physics Communications 197, 136–143. doi: 10.1016/j.cpc.2015.08.025.
Open DOI Search in Google Scholar Back to article
Bastin, G. and Coron, J.-M. (2016) Stability and Boundary Stabilization of 1-D Hyperbolic Systems, 88. Progress in Nonlinear Differential Equations and their Applications. Subseries in Control. Birkhäuser/Springer, doi: 10.1007/978-3-319-32062-5.
Open DOI Search in Google Scholar Back to article
Benamou, J.-D. (1996) A domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations. SIAM Journal on Numerical Analysis 33(6), 2401–2416. doi: 10.1137/S0036142994267102.
Open DOI Search in Google Scholar Back to article
Casas, E. and Fernández, L. A. (1989) A GreenâĂŹs formula for quasilinear elliptic operators. Journal of Mathematical Analysis and Applications 142(1), 62–73. doi: 10.1016/0022-247X(89)90164-9.
Open DOI Search in Google Scholar Back to article
Gander, M. J. and Kwok, F. (2016) Schwarz methods for the time-parallel solution of parabolic control problems. Domain Decomposition Methods in Science and Engineering XXII. Lecture Notes in Computational Science and Engineering, 104. Springer, Cham, 207–216. doi: 10.1007/978-3-319-18827-0_19.
Open DOI Search in Google Scholar Back to article
Gander, M. J., Kwok, F. and Salomon, J. (2020) ParaOpt: a parareal algorithm for optimality systems. SIAM Journal on Scientific Computing 42(5), A2773–A2802. doi: 10.1137/19M1292291.
Open DOI Search in Google Scholar Back to article
Gander, M. J. and Vandewalle, S. (2007) Analysis of the parareal time-parallel time-integration method. SIAM Journal on Scientific Computing 29(2), 556–578. doi: 10.1137/05064607X.
Open DOI Search in Google Scholar Back to article
Glowinski, R. and Le Tallec, P. (1990) Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method. In: Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989). SIAM, Philadelphia, PA, 224–231.
Search in Google Scholar Back to article
Gugat, M. (2021) On the turnpike property with interior decay for optimal control problems. Mathematics of Control, Signals, and Systems 33(2), 237–258. doi: 10.1007/s00498-021-00280-4.
Open DOI Search in Google Scholar Back to article
Gugat, M., Giesselmann, J. and Kunkel, T. (2021) Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks. IMA Journal of Mathematical Control and Information (Sept. 2021). doi: 10.1093/imamci/dnab029.
Open DOI Search in Google Scholar Back to article
Hante, F. M., Leugering, G., Martin, A., Schewe, L. and Schmidt, M. (2017) Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications. In: P. Manchanda, R. Lozi and A. H. Siddiqi, eds., Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms. Industrial and Applied Mathematics, 77–122. doi: 10.1007/978-981-10-3758-0_5.
Open DOI Search in Google Scholar Back to article
Heinkenschloss, M. (2005) A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems. Journal of Computational and Applied Mathematics 173(1), 169–198. doi: 10.1016/j.cam.2004.03.005.
Open DOI Search in Google Scholar Back to article
Krug, R., Leugering, G., Martin, A., Schmidt, M. and Weninger, D. (2021) Time-Domain Decomposition for Optimal Control Problems Governed by Semi-linear Hyperbolic Systems. SIAM Journal on Control and Optimization. url: http://www.optimization-online.org/DB_HTML/2020/11/8132.html. Forthcoming.
Search in Google Scholar Back to article
Lagnese, J. E. and Leugering, G. (2002) Time domain decomposition in final value optimal control of the Maxwell system. ESAIM: Control, Optimisation and Calculus of Variations 8, 775–799. doi: 10.1051/cocv:2002042.
Open DOI Search in Google Scholar Back to article
Lagnese, J. E. and Leugering, G. (2003) Time-domain decomposition of optimal control problems for the wave equation. Systems & Control Letters 48(3-4), 229–242. doi: 10.1016/S0167-6911(02)00268-2.
Open DOI Search in Google Scholar Back to article
Lagnese, J. E. and Leugering, G. (2004) Domain decomposition methods in optimal control of partial differential equations. International Series of Numerical Mathematics 148. Birkhäuser Verlag, Basel. doi: 10.1007/978-3-0348-7885-2.
Open DOI Search in Google Scholar Back to article
Leugering, G., Martin, A., Schmidt, M. and Sirvent, M. (2017) Nonoverlapping domain decomposition for optimal control problems governed by semilinear models for gas flow in networks. Control and Cybernetics 46(3), 191–225.
Search in Google Scholar Back to article
Leugering, G. and Rodriguez, C. (2020) Boundary feedback stabilization for the intrinsic geometrically exact beam model. Technical report. url: https://arxiv.org/abs/1912.02543.
Search in Google Scholar Back to article
Leugering, G. and Schmidt, J. G. (2002) On the modelling and stabilization of flows in networks of open canals. SIAM Journal on Control and Optimization 41(1), 164–180. doi: 10.1137/S0363012900375664.
Open DOI Search in Google Scholar Back to article
Lions, J.-L. (1990) On the Schwarz alternating method. III. A variant for nonover-lapping subdomains. In: Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989). SIAM, Philadelphia, PA, 202–223.
Search in Google Scholar Back to article
Lions, J.-L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Die Grundlehren der mathematischen Wissenschaften, Band 170. Translated from French by S. K. Mitter. Springer-Verlag, New York-Berlin.10.1007/978-3-642-65024-6_4
Search in Google Scholar Back to article
Lions, J.-L, Maday, Y. and Turinici, G. (2001) Résolution d’EDP par un schéma en temps “pararéel”. Comptes Rendus de l’Académie des Sciences - Series I -Mathematics 332(7), 661–668. doi: 10.1016/S0764-4442(00)01793-6.
Open DOI Search in Google Scholar Back to article
Maday, Y., Salomon, J. and Turinici, G. (2006) Monotonic time-discretized schemes in quantum control. Numerische Mathematik 103(2), 323–338. doi: 10.1007/s00211-006-0678-x.
Open DOI Search in Google Scholar Back to article
Maday, Y. and Turinici, G. (2002) A parareal in time procedure for the control of partial differential equations. Comptes Rendus Mathématique. Académie des Sciences. Paris 335(4), 387–392. doi: 10.1016/S1631-073X(02)02467-6.
Open DOI Search in Google Scholar Back to article
Prodi, G., ed. (2010) Problems in Non-Linear Analysis. Centro Internazionale Matematico Estivo (C.I.M.E.) 55 Summer Schools. Springer, Heidelberg; Fondazione C.I.M.E., Florence.
Search in Google Scholar Back to article
Roubicek, T. (2005) Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, 153. Birkhäuser Verlag, Basel. doi: 10.1007/978-3-0348-0513-1.
Open DOI Search in Google Scholar Back to article
Schaefer, H. (1957) Über die Methode sukzessiver Approximationen. Jahresbericht der Deutschen Mathematiker-Vereinigung 59, 131–140.
Search in Google Scholar Back to article
Ulbrich, S. (2007) Generalized SQP methods with “parareal” time-domain decomposition for time-dependent PDE-constrained optimization. In: Real-time PDE-constrained Optimization. Computational Science & Engineering 3, SIAM, 145–168. doi: 10.1137/1.9780898718935.ch7.
Open DOI Search in Google Scholar Back to article
Vrabie, I. I. (2003) C0-Semigroups and Applications. North-Holland Mathematics Studies 191. North-Holland Publishing Co., Amsterdam.
Search in Google Scholar Back to article
Wu, S.-L. and Huang, T.-Z. (2018) A fast second-order parareal solver for fractional optimal control problems. Journal of Vibration and Control 24(15), 3418–3433. doi: 10.1177/1077546317705557.
Open DOI Search in Google Scholar Back to article
Wu, S.-L. and Liu, J. (2020) A parallel-in-time block-circulant preconditioner for optimal control of wave equations. SIAM Journal on Scientific Computing 42(3), A1510–A1540. doi: 10.1137/19M1289613.
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/candc-2021-0026 | Journal eISSN: 2720-4278 | Journal ISSN: 0324-8569

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Language: English

Page range: 427 - 455

Submitted on: Jul 1, 2021

Accepted on: Nov 1, 2021

Published on: Jun 27, 2022

Published by: Systems Research Institute Polish Academy of Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

time-domain decomposition,

optimal control,

semilinear hyperbolic systems,

convergence

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Electrical engineering,

Fundamentals of electrical engineering,

Mechanical engineering,

Fundamentals of mechanical engineering,

Mathematics,

General mathematics

© 2022 Richard Krug, Günter Leugering, Alexander Martin, Martin Schmidt, Dieter Weninger, published by Systems Research Institute Polish Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 50 (2021): Issue 4 (December 2021)