Control problems on infinite horizon subject to time-dependent pure state constraints
In the last decades, control problems with infinite horizons and discount factors have become increasingly central not only for economics but also for applications in artificial intelligence and machine learning. The strong links between reinforcement learning and control theory have led to major ef...
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| Published in | Mathematics of control, signals, and systems Vol. 36; no. 2; pp. 423 - 450 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
London
Springer London
01.06.2024
Springer Nature B.V |
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| Abstract | In the last decades, control problems with infinite horizons and discount factors have become increasingly central not only for economics but also for applications in artificial intelligence and machine learning. The strong links between reinforcement learning and control theory have led to major efforts toward the development of algorithms to learn how to solve constrained control problems. In particular, discount plays a role in addressing the challenges that come with models that have unbounded disturbances. Although algorithms have been extensively explored, few results take into account time-dependent state constraints, which are imposed in most real-world control applications. For this purpose, here we investigate feasibility and sufficient conditions for Lipschitz regularity of the value function for a class of discounted infinite horizon optimal control problems subject to time-dependent constraints. We focus on problems with data that allow nonautonomous dynamics, and Lagrangian and state constraints that can be unbounded with possibly nonsmooth boundaries. |
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| AbstractList | In the last decades, control problems with infinite horizons and discount factors have become increasingly central not only for economics but also for applications in artificial intelligence and machine learning. The strong links between reinforcement learning and control theory have led to major efforts toward the development of algorithms to learn how to solve constrained control problems. In particular, discount plays a role in addressing the challenges that come with models that have unbounded disturbances. Although algorithms have been extensively explored, few results take into account time-dependent state constraints, which are imposed in most real-world control applications. For this purpose, here we investigate feasibility and sufficient conditions for Lipschitz regularity of the value function for a class of discounted infinite horizon optimal control problems subject to time-dependent constraints. We focus on problems with data that allow nonautonomous dynamics, and Lagrangian and state constraints that can be unbounded with possibly nonsmooth boundaries. |
| Author | Basco, Vincenzo |
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| CitedBy_id | crossref_primary_10_1007_s40574_024_00410_1 |
| Cites_doi | 10.1090/S0002-9947-1983-0690039-8 10.1016/j.ifacol.2017.08.979 10.1007/978-0-8176-4848-0 10.1016/j.jet.2016.08.001 10.1109/CDC.2016.7798271 10.1016/j.automatica.2014.10.096 10.3934/mcrf.2018022 10.1109/TAC.2010.2086553 10.1109/TAC.2005.861702 10.1007/978-3-031-22216-0_12 10.1007/978-3-030-17949-6_2 10.1090/S0002-9947-1984-0732102-X 10.1007/978-1-4471-5076-3 10.1109/CDC.2013.6761117 10.1016/j.automatica.2010.06.034 10.1007/s10479-018-2947-3 10.1016/B978-1-55860-377-6.50013-X 10.1109/CDC.2013.6760907 10.1016/j.jmaa.2022.126452 10.1214/aoms/1177700285 10.1007/978-3-319-75169-6 10.1016/B978-1-55860-377-6.50040-2 10.1007/s00030-019-0553-y 10.1109/TAC.2014.2303232 10.1109/TAC.2013.2270052 10.1109/TAC.2012.2203054 10.1109/CDC.2018.8619554 10.2514/3.25375 10.1007/BFb0026710 10.1023/A:1007686309208 10.1109/TAC.2016.2616644 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Keywords | Viability Regularity of value functions Infinite horizon control problems State constraints |
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| References | De Jager B, Van Keulen T (2013) Optimal control of hybrid vehicles. Springer, Kessels KouvaritakisBCannonMCouchmanPMpc as a tool for sustainable development integrated policy assessmentIEEE Trans Autom Control2006511145149219280310.1109/TAC.2005.861702 FrankelADiscounted quotasJ Econ Theory2016166396444356645010.1016/j.jet.2016.08.001 MunosRA study of reinforcement learning in the continuous case by the means of viscosity solutionsMach Learn20004026529910.1023/A:1007686309208 Basco V, Frankowska H (2019) Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints. In: Alabau-Boussouira F, et al (eds) Trends in control theory and partial differential equations, vol 32 of Springer INdAM Series. Springer International Publishing, pp 15 – 52 Gordon GJ (1995) Stable function approximation in dynamic programming. In: Machine learning proceedings. Elsevier, pp 261–268 KamgarpourMSummersTOn infinite dimensional linear programming approach to stochastic controlIFAC-PapersOnLine20175016148615310.1016/j.ifacol.2017.08.979 SchildbachGGoulartPMorariMLinear controller design for chance constrained systemsAutomatica201551278284328477910.1016/j.automatica.2014.10.096 Hashimoto T (2013) Probabilistic constrained model predictive control for linear discrete-time systems with additive stochastic disturbances. In: 52nd IEEE conference on decision and control. IEEE, pp 6434–6439 Bertsekas D (2022) Dynamic programming and optimal control, volume 1. Athena scientific Aubin J-P, Frankowska H (2009) Set-valued analysis. Modern Birkhäuser Classics. Birkhäuser Boston Inc, Boston, MA CannonMKouvaritakisBRakovićSVChengQStochastic tubes in model predictive control with probabilistic constraintsIEEE Trans Autom Control2010561194200277721810.1109/TAC.2010.2086553 BlackwellDDiscounted dynamic programmingAnn Math Stat196536122623517353610.1214/aoms/1177700285 Bertsekas D (2019) Reinforcement learning and optimal control. Athena Scientific Farina M, Giulioni L, Magni L, Scattolini R (2013) A probabilistic approach to model predictive control. In: 52nd IEEE conference on decision and control. IEEE, pp 7734–7739 CrandallMGLionsP-LViscosity solutions of Hamilton–Jacobi equationsTrans Amer Math Soc1983277114269003910.1090/S0002-9947-1983-0690039-8 PostoyanRBuşoniuLNešićDDaafouzJStability analysis of discrete-time infinite-horizon optimal control with discounted costIEEE Trans Autom Control201662627362749366055910.1109/TAC.2016.2616644 Baird L (1995) Residual algorithms: reinforcement learning with function approximation. In: Machine learning proceedings. Elsevier, pp 30–37 KouvaritakisBCannonMRakovićSVChengQExplicit use of probabilistic distributions in linear predictive controlAutomatica2010461017191724287732810.1016/j.automatica.2010.06.034 Hewing L, Zeilinger MN (2018) Stochastic model predictive control for linear systems using probabilistic reachable sets. In: 2018 IEEE conference on decision and control (CDC), pp 5182–5188. IEEE VinterRBOptimal Control2000Boston, MABirkhäuser MargellosKGoulartPLygerosJOn the road between robust optimization and the scenario approach for chance constrained optimization problemsIEEE Trans Autom Control201459822582263324527110.1109/TAC.2014.2303232 BascoVCannarsaPFrankowskaHNecessary conditions for infinite horizon optimal control problems with state constraintsMath Control Relat Fields201883–4535555391745210.3934/mcrf.2018022 BascoVFrankowskaHHamilton–Jacobi–Bellman equations with time-measurable data and infinite horizonNonlinear Differ Equ Appl20192617390794710.1007/s00030-019-0553-y CrandallMGEvansLCLionsP-LSome properties of viscosity solutions of Hamilton–Jacobi equationsTrans Amer Math Soc1984282248750273210210.1090/S0002-9947-1984-0732102-X Munos R (1998) A general convergence method for reinforcement learning in the continuous case. In: European conference on machine learning. 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IEEE MenonPKABriggsMMNear-optimal midcourse guidance for air-to-air missilesJ Guid Control Dyn199013459660210.2514/3.25375 NystrupPBoydSLindströmEMadsenHMulti-period portfolio selection with drawdown controlAnn Oper Res20192821245271401923910.1007/s10479-018-2947-3 CalafioreGCFagianoLRobust model predictive control via scenario optimizationIEEE Trans Autom Control2012581219224300672510.1109/TAC.2012.2203054 Weston J, Tolić D, Palunko I (2022) Mixed use of Pontryagin’s principle and the Hamilton–Jacobi–Bellman equation in infinite-and finite-horizon constrained optimal control. In: International conference on intelligent autonomous systems. Springer, pp 167–185 Van ParysBPGGoulartPJMorariMInfinite horizon performance bounds for uncertain constrained systemsIEEE Trans Autom Control2013581128032817312599010.1109/TAC.2013.2270052 372_CR21 372_CR20 D Blackwell (372_CR9) 1965; 36 BPG Van Parys (372_CR33) 2013; 58 GC Calafiore (372_CR11) 2012; 58 MG Crandall (372_CR13) 1984; 282 372_CR7 372_CR6 372_CR1 R Munos (372_CR29) 2000; 40 G Schildbach (372_CR32) 2015; 51 372_CR2 372_CR18 372_CR17 RB Vinter (372_CR34) 2000 372_CR16 372_CR15 B Kouvaritakis (372_CR25) 2010; 46 372_CR8 372_CR35 M Kamgarpour (372_CR23) 2017; 50 PKA Menon (372_CR27) 1990; 13 372_CR10 MG Crandall (372_CR14) 1983; 277 B Kouvaritakis (372_CR24) 2006; 51 M Cannon (372_CR12) 2010; 56 V Basco (372_CR4) 2018; 8 P Nystrup (372_CR30) 2019; 282 K Margellos (372_CR26) 2014; 59 V Basco (372_CR3) 2022; 515 R Postoyan (372_CR31) 2016; 62 V Basco (372_CR5) 2019; 26 372_CR28 A Frankel (372_CR19) 2016; 166 372_CR22 |
| References_xml | – reference: Bertsekas D (2019) Reinforcement learning and optimal control. Athena Scientific – reference: Basco V, Frankowska H (2019) Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints. In: Alabau-Boussouira F, et al (eds) Trends in control theory and partial differential equations, vol 32 of Springer INdAM Series. Springer International Publishing, pp 15 – 52 – reference: Munos R (1998) A general convergence method for reinforcement learning in the continuous case. In: European conference on machine learning. Springer, pp 394–405 – reference: CrandallMGLionsP-LViscosity solutions of Hamilton–Jacobi equationsTrans Amer Math Soc1983277114269003910.1090/S0002-9947-1983-0690039-8 – reference: CannonMKouvaritakisBRakovićSVChengQStochastic tubes in model predictive control with probabilistic constraintsIEEE Trans Autom Control2010561194200277721810.1109/TAC.2010.2086553 – reference: Feichtinger G, Kovacevic RM, Tragler G (2018) Control systems and mathematical methods in economics, volume 687. Lecture Notes in Economics and Mathematical Systems. Springer – reference: Boyan J, Moore A (1994) Generalization in reinforcement learning: safely approximating the value function. Adv Neural Inf Process Syst 7 – reference: BascoVCannarsaPFrankowskaHNecessary conditions for infinite horizon optimal control problems with state constraintsMath Control Relat Fields201883–4535555391745210.3934/mcrf.2018022 – reference: BascoVFrankowskaHHamilton–Jacobi–Bellman equations with time-measurable data and infinite horizonNonlinear Differ Equ Appl20192617390794710.1007/s00030-019-0553-y – reference: FrankelADiscounted quotasJ Econ Theory2016166396444356645010.1016/j.jet.2016.08.001 – reference: Bertsekas D (2022) Dynamic programming and optimal control, volume 1. Athena scientific – reference: CalafioreGCFagianoLRobust model predictive control via scenario optimizationIEEE Trans Autom Control2012581219224300672510.1109/TAC.2012.2203054 – reference: BascoVWeak epigraphical solutions to Hamilton–Jacobi–Bellman equations on infinite horizonJ Math Anal Appl20225152444618310.1016/j.jmaa.2022.126452 – reference: Farina M, Giulioni L, Magni L, Scattolini R (2013) A probabilistic approach to model predictive control. In: 52nd IEEE conference on decision and control. IEEE, pp 7734–7739 – reference: KamgarpourMSummersTOn infinite dimensional linear programming approach to stochastic controlIFAC-PapersOnLine20175016148615310.1016/j.ifacol.2017.08.979 – reference: KouvaritakisBCannonMRakovićSVChengQExplicit use of probabilistic distributions in linear predictive controlAutomatica2010461017191724287732810.1016/j.automatica.2010.06.034 – reference: Weston J, Tolić D, Palunko I (2022) Mixed use of Pontryagin’s principle and the Hamilton–Jacobi–Bellman equation in infinite-and finite-horizon constrained optimal control. In: International conference on intelligent autonomous systems. Springer, pp 167–185 – reference: De Jager B, Van Keulen T (2013) Optimal control of hybrid vehicles. Springer, Kessels – reference: De Pinho MR, Foroozandeh Z, Matos A (2016) Optimal control problems for path planing of AUV using simplified models. In: 2016 IEEE 55th conference on decision and control (CDC), pp 210–215. IEEE – reference: Baird L (1995) Residual algorithms: reinforcement learning with function approximation. In: Machine learning proceedings. Elsevier, pp 30–37 – reference: KouvaritakisBCannonMCouchmanPMpc as a tool for sustainable development integrated policy assessmentIEEE Trans Autom Control2006511145149219280310.1109/TAC.2005.861702 – reference: Hewing L, Zeilinger MN (2018) Stochastic model predictive control for linear systems using probabilistic reachable sets. In: 2018 IEEE conference on decision and control (CDC), pp 5182–5188. IEEE – reference: BlackwellDDiscounted dynamic programmingAnn Math Stat196536122623517353610.1214/aoms/1177700285 – reference: VinterRBOptimal Control2000Boston, MABirkhäuser – reference: CrandallMGEvansLCLionsP-LSome properties of viscosity solutions of Hamilton–Jacobi equationsTrans Amer Math Soc1984282248750273210210.1090/S0002-9947-1984-0732102-X – reference: Gordon GJ (1995) Stable function approximation in dynamic programming. 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| SubjectTerms | Algorithms Artificial intelligence Communications Engineering Constraints Control Control theory Discounts Machine learning Mathematics Mathematics and Statistics Mechatronics Networks Optimal control Original Article Robotics Systems Theory Time dependence |
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| Title | Control problems on infinite horizon subject to time-dependent pure state constraints |
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