Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is writt...

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Veröffentlicht in:	Computational optimization and applications Jg. 77; H. 1; S. 307 - 334
Hauptverfasser:	Delanoue, Nicolas, Lhommeau, Mehdi, Lagrange, Sébastien
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.09.2020 Springer Nature B.V Springer Verlag
Schlagworte:	Convex and Discrete Geometry Fineness Interval arithmetic Linear programming Lower bounds Management Science Mathematical programming Mathematics Mathematics and Statistics Nonlinear control Operations Research Operations Research/Decision Theory Optimal control Optimization Optimization and Control Statistics Nonlinear optimal control Interval arithmetic Continuous programming Optimization
ISSN:	0926-6003, 1573-2894
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Abstract	This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.
AbstractList	This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.
Author	Lhommeau, Mehdi Lagrange, Sébastien Delanoue, Nicolas
Author_xml	– sequence: 1 givenname: Nicolas orcidid: 0000-0001-6927-6281 surname: Delanoue fullname: Delanoue, Nicolas email: nicolas.delanoue@univ-angers.fr organization: LARIS, Université d’Angers – sequence: 2 givenname: Mehdi surname: Lhommeau fullname: Lhommeau, Mehdi organization: LARIS, Université d’Angers – sequence: 3 givenname: Sébastien surname: Lagrange fullname: Lagrange, Sébastien organization: LARIS, Université d’Angers
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Cites_doi	10.1137/040616802 10.1137/1030153 10.1093/imamci/17.2.167 10.1073/pnas.38.8.716 10.1090/S0002-9947-1983-0690039-8 10.1016/S0362-546X(00)85016-6 10.1137/1.9781611973051 10.1090/gsm/058 10.1109/CDC.2008.4739136 10.1090/S0025-5718-1984-0744921-8 10.1007/s10589-015-9794-9 10.1137/1.9780898718577 10.1137/S1052623497315768 10.1007/s00245-014-9257-1 10.1137/070685051 10.1137/0331024 10.1007/978-1-4471-4820-3 10.1090/S0002-9947-1984-0732102-X 10.1090/S0273-0979-1992-00266-5 10.1137/060655286
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Keywords	Nonlinear optimal control Interval arithmetic Continuous programming Optimization
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References	The gnu multiple precision arithmetic library. https://gmplib.org Hernández-HernándezDHernández-LermaOTaksarMThe linear programming approach to deterministic optimal control problemsAppl. Math.1996241173314049810871.49005 CrandallMGLionsPLTwo approximations of solutions of Hamilton–Jacobi equationsMath. Comput.19844316711974492110.1090/S0025-5718-1984-0744921-8 High performance computing cluster of leria.: Slurm/debian cluster of 27 nodes(700 logical CPU, 2 nvidia GPU tesla k20m, 1 nvidia P100 GPU), 120TB of beegfs scratch storage (2018) CrandallMGIshiiHLionsPLUser’s guide to viscosity solutions of second order partial differential equations.Bull. Am. Math. Soc. New Ser.1992271167111869910.1090/S0273-0979-1992-00266-50755.35015 Filib++ interval library. http://www2.math.uni-wuppertal.de/~xsc/software/filib.html JaulinLKiefferMDidritOWalterEApplied Interval Analysis: With Examples in Parameter and State Estimation. Robust Control and Robotics.2012LondonSpringer1023.65037 GaitsgoryVRossomakhineSLinear programming approach to deterministic long run average problems of optimal controlSIAM J. Control Optim.200644620062037224817310.1137/0406168021109.93017 GaitsgoryVRossomakhineSAveraging and linear programming in some singularly perturbed problems of optimal controlAppl. Math. Optim.2015712195276332358910.1007/s00245-014-9257-11317.49042 ClarkeFFunctional Analysis. Calculus of Variations and Optimal Control. Graduate Texts in Mathematics.2013LondonSpringer10.1007/978-1-4471-4820-3 LasserreJMoments. Positive Polynomials and Their Applications: Imperial College Press Optimization Series2010LondonImperial College Press WangSGaoFTeoKAn upwind finite-difference method for the approximation of viscosity solutions to Hamilton–Jacobi–Bellman equationsIMA J. Math. Control Inf.2000172167178176927410.1093/imamci/17.2.167 RudinWReal and Complex Analysis19873New YorkMcGraw-Hill Book Co.0925.00005 JungeOOsingaHMA set oriented approach to global optimal controlESAIM: Control Optim. Calc. Var.200410225927020834871072.49014 Hairer, E., Nørsett, S., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Lecture Notes in Economic and Mathematical Systems. Springer, Berlin (1993) CrandallMGLionsPLViscosity solutions of Hamilton–Jacobi equationsTrans. Am. Math. Soc.1983277114269003910.1090/S0002-9947-1983-0690039-8 Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton–Jacobi Equations. SIAM (2014). https://hal.inria.fr/hal-00916055 Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I (2nd Revised. Ed.): Nonstiff Problems. 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Theory Methods Appl.2000401–8279293176841310.1016/S0362-546X(00)85016-6 VinterRConvex duality and nonlinear optimal controlSIAM J. Control Optim.1993312518538120598710.1137/03310240781.49012 Evans, L.: An Introduction to Mathematical Optimal Control Theory, version 0.2. http://math.berkeley.edu/~evans MooreRInterval Analysis. Prentice-Hall Series in Automatic Computation1966Upper Saddle RiverPrentice-Hall DelanoueNLhommeauMLucidarmePNumerical enclosures of the optimal cost of the kantorovitch’s mass transportation problemComput. Optim. Appl.2015111910.1007/s10589-015-9794-91364.90211 BalderEControl and optimisation: the linear treatment of nonlinear problems (J. E. Rubio)SIAM Rev.198830466366310.1137/1030153 BellmanROn the theory of dynamic programmingProc. Natl. Acad. 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References_xml	– reference: CrandallMGIshiiHLionsPLUser’s guide to viscosity solutions of second order partial differential equations.Bull. Am. Math. Soc. New Ser.1992271167111869910.1090/S0273-0979-1992-00266-50755.35015 – reference: DelanoueNLhommeauMLucidarmePNumerical enclosures of the optimal cost of the kantorovitch’s mass transportation problemComput. Optim. Appl.2015111910.1007/s10589-015-9794-91364.90211 – reference: Glpk: Gnu linear programming kit. http://www.gnu.org/software/glpk/ – reference: HuangCSWangSTeoKSolving Hamilton–Jacobi–Bellman equations by a modified method of characteristicsNonlinear Anal. Theory Methods Appl.2000401–8279293176841310.1016/S0362-546X(00)85016-6 – reference: VinterRConvex duality and nonlinear optimal controlSIAM J. Control Optim.1993312518538120598710.1137/03310240781.49012 – reference: CrandallMGLionsPLTwo approximations of solutions of Hamilton–Jacobi equationsMath. Comput.19844316711974492110.1090/S0025-5718-1984-0744921-8 – reference: NeumaierAInterval Methods for Systems of Equations1990CambridgeCambridge University Press, Cambridge Middle East Library0715.65030 – reference: RubioJControl and Optimization: The Linear Treatment of Nonlinear Problems1986ManchesterManchester University Press1095.49500 – reference: The gnu multiple precision arithmetic library. https://gmplib.org/ – reference: GaitsgoryVRossomakhineSAveraging and linear programming in some singularly perturbed problems of optimal controlAppl. Math. Optim.2015712195276332358910.1007/s00245-014-9257-11317.49042 – reference: MooreRInterval Analysis. Prentice-Hall Series in Automatic Computation1966Upper Saddle RiverPrentice-Hall – reference: RudinWReal and Complex Analysis19873New YorkMcGraw-Hill Book Co.0925.00005 – reference: VillaniCTopics in Optimal Transportation: Graduate Studies in Mathematics2003ProvidenceAmerican Mathematical Society10.1090/gsm/058 – reference: AkianMGaubertSLakhouaAThe max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysisSIAM J. Control Optim.2008472817848238586410.1137/0606552861157.49034 – reference: LasserreJBHenrionDPrieurCTrélatENonlinear optimal control via occupation measures and lmi-relaxationsSIAM J. Control Optim.200847416431666242132410.1137/0706850511188.90193 – reference: High performance computing cluster of leria.: Slurm/debian cluster of 27 nodes(700 logical CPU, 2 nvidia GPU tesla k20m, 1 nvidia P100 GPU), 120TB of beegfs scratch storage (2018) – reference: ClarkeFFunctional Analysis. Calculus of Variations and Optimal Control. Graduate Texts in Mathematics.2013LondonSpringer10.1007/978-1-4471-4820-3 – reference: FalconeMNumerical solution of dynamic programming equations: optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations1997BaselBirkhäuser – reference: Evans, L.: An Introduction to Mathematical Optimal Control Theory, version 0.2. http://math.berkeley.edu/~evans/ – reference: WangSGaoFTeoKAn upwind finite-difference method for the approximation of viscosity solutions to Hamilton–Jacobi–Bellman equationsIMA J. Math. Control Inf.2000172167178176927410.1093/imamci/17.2.167 – reference: BellmanROn the theory of dynamic programmingProc. Natl. Acad. Sci.19523887167195085610.1073/pnas.38.8.716 – reference: Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I (2nd Revised. Ed.): Nonstiff Problems. 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Snippet	This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two...
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SubjectTerms	Convex and Discrete Geometry Fineness Interval arithmetic Linear programming Lower bounds Management Science Mathematical programming Mathematics Mathematics and Statistics Nonlinear control Operations Research Operations Research/Decision Theory Optimal control Optimization Optimization and Control Statistics
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Title	Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis
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