Space-Time-Domain Decomposition for Optimal Control Problems Governed by Linear Hyperbolic Systems

G¨unter Leugering

Анотація


In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work [31, 32] and answer a long standing open question as to whether the combination of time- and space-domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by P. L. Lions [25] for elliptic problems.

Ключові слова


space- and time-domain decomposition; optimal control; linear hyperbolic systems; convergence; a posteriori error estimates

Повний текст:

PDF (English)

Посилання


M. Brokate, Necessary optimality conditions for the control of semilinear hyperbolic boundary value problems, SIAM Journal on Control and Optimization, 25(5) (1985), 1353–1369.

J. E. Lagnese, G. Leugering, Time-domain decomposition of optimal control problems for the wave equation, Systems & Control Letters, 48(3-4) (2003), 229–242.

J. E. Lagnese, G. Leugering, Domain decomposition methods in optimal control of partial differential equations, International Series of Numerical Mathematics, Birkh¨auser-Verlag, 148, 2004.

Lagnese, John E and Leugering, G¨unter,Time domain decomposition in final value optimal control of the Maxwell system, ESAIM: Control, Optimisation and Calculus of Variations 8 (2002), 775–799.

H. Schaefer, ¨ Uber die Methode sukzessiver Approximationen, Jahresbericht der Deutschen Mathematiker-Vereinigung, 59 (1957), 131–140.

G. Bastin, J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, Birkh¨auser/Springer, Progress in Nonlinear Differential Equations and their Applications, Subseries in Control, 88 (2016).

J.-L. Lions, Y. Maday, G. Turinici, Resolution d’EDP par un schema en temps ‘parareel’, Comptes Rendus de l’Academie des Sciences, Series I. Mathematics, 332(7) (2001), 661–668.

M. J. Gander, S. Vandewalle, Analysis of the parareal time-parallel timeintegration method, SIAM Journal on Scientific Computing, 29(2) (2007), 556–578.

S. Ulbrich, Generalized SQP methods with ‘parareal’ time-domain decomposition for time-dependent PDE-constrained optimization, SIAM, Real-time PDEconstrained optimization,Computational Science & Engineering, 3 (2007), 145–168.

Gander, Martin J., Analysis of the parareal algorithm applied to hyperbolic problems using characteristics, Boletin de la Sociedad Espanola de Matematica Aplicada, 42 (2008), 21–35.

M. Heinkenschloss, A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems, Journal of Computational and Applied Mathematics, 173(1) (2005), 169–198.

W. Schmaedeke, Mathematical theory of optimal control for semilinear hyperbolic systems in two independent variables, SIAM Journal on Control, 5 (1967), 138–153.

G. Leugering, J. P. G. Schmidt, On the modelling and stabilization of flows in networks of open canals, SIAM Journal on Control and Optimization, 41(1) (2002), 164–180.

G. Leugering, C. Rodriguez, Boundary feedback stabilization for the intrinsic geometrically exact beam model, https://arxiv.org/abs/1912.02543, (2020).

G. Leugering, A. Martin, M. Schmidt, M. Sirvent, Nonoverlapping domain decomposition for optimal control problems governed by semilinear models for gas flow in networks, Control and Cybernetics, 46(3) (2017), 191–225.

F. M. Hante, G. Leugering, A. Martin, L. Schewe, M. Schmidt, Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications, Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms, Manchanda, Pammy and Lozi, Rene and Siddiqi, Abul Hasan, Eds.,Industrial and Applied Mathematics, (2017), 77–122.

M. J. Gander, F. Kwok, J. Salomon, ParaOpt: a parareal algorithm for optimality systems, SIAM Journal on Scientific Computing, 42(5) (2020), A2773–A2802.

M. J. Gander, F. Kwok, 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, Springer, Cham, 104 (2016), 207–216.

Y. Maday, M.-K. Riahi, J. Salomon, Parareal in time intermediate targets methods for optimal control problems, Control and optimization with PDE constraints, Internat. Ser. Numer. Math., 164, 79–92, Birkh¨auser/Springer Basel AG, Basel, 2013.

A. T. Barker, M. Stoll, Domain decomposition in time for PDE-constrained optimization, Computer Physics Communications, 197 (2015), 136–143.

Wu, Shu-Lin and Huang, Ting-Zhu, A fast second-order parareal solver for fractional optimal control problems, Journal of Vibration and Control, 24(15) (2018), 3418–3433.

S.-L. Wu, J. Liu, A parallel-in-time block-circulant preconditioner for optimal control of wave equations, SIAM Journal on Scientific Computing, 42(3) (2020), A1510–A1540.

Y. Maday, J. Salomon, G. Turinici, Monotonic time-discretized schemes in quantum control, Numerische Mathematik, 103(2) (2006), 323–338.

Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations, Comptes Rendus Math´ematique. Acad emie des Sciences, Paris, 335(4) (2002), 387–392.

P.-L. Lions, On the Schwarz alternating method. III. A variant for nonoverlapping subdomains, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), 202–223, SIAM, Philadelphia, PA, 1989.

R. Glowinski, P. Le Tallec, Augmented Lagrangian interpretation of the nonoverlapping Schwarz alternating method, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston,TX, 1989), 224–231, SIAM, Philadelphia, PA, 1990.

H. Bialy, Iterative Behandlung linearer Funktionalgleichungen, Archive for Rational Mechanics and Analysis, 4 (1959), 166–176.

M. J. Gander, L. Halpern, F. Nataf, Domain decomposition methods for wave propagation, Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 807–811.

F.-C. Otto, G. Lube, A posteriori estimates for a non-overlapping domain decomposition method, Computing, 62 (1) (1999), 27–43.

S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 1 (2011), 324–342.

R. Krug, G. Leugering, A. Martin, M. Schmidt, D. Weninger, Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems with Mixed Two-Point Boundary Conditions (submitted), Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, 20, (2021).

R. Krug, G. Leugering, A. Martin, M. Schmidt, D. Weninger, Time-Domain Decomposition for Optimal Control Problems Governed by Semilinear Hyperbolic Systems (to appear in SICON), Friedrich-Alexander- Universit¨at Erlangen-N¨urnberg, 28, (2021).

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73, (1967), 591–597.

C. Corduneanu, Integral equations and applications, Cambridge University Press, Cambridge, (1991).

D. L. Russell, Optimal regulation of linear symmetric hyperbolic systems with finite dimensional controls, SIAM J. Control, 4, (1966), 276–294.




DOI: http://dx.doi.org/10.15421/142107

Посилання

  • Поки немає зовнішніх посилань.



Індексування журналу

Журнал розміщено у наукометричних базах, репозитаріях та пошукових системах:

                 


Адреса редколегії: 49050, Україна, Дніпровський національний університет імені Олеся Гончара, вул. Козакова 18, корп. 14, механіко-математичний факультет, д-р фіз.-мат. наук, проф. Когут П.І. 

email: p.kogut@i.ua

www.dnu.dp.ua


Free counters! 

Лицензия Creative Commons
Це видання має доступ за ліцензією Creative Commons «Attribution» («Атрибуция») 4.0 Всемирная.


Open Science in Ukraine - website development