Error estimates for second order Hamilton-Jacobi-Bellman equations.
Approximation of probabilistic reachable sets

Mohamed Assellaou; Olivier Bokanowski; Hasnaa Zidani

doi:10.3934/dcds.2015.35.3933

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September 2015, 35(9): 3933-3964. doi: 10.3934/dcds.2015.35.3933

Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets

Mohamed Assellaou ^1, , Olivier Bokanowski ^2, and Hasnaa Zidani ^1,

1.	Unité des mathématiques appliquées (UMA), ENSTA ParisTech, 828 Bd Maréchaux, 91120 Palaiseau, France, France
2.	Laboratoire Jacques-Louis Lions, UMR 7598, Université Paris-Diderot (Paris 7), UFR de Mathématiques - 5 rue Thomas Mann, 75205 Paris CEDEX 13, France

Received May 2014 Revised November 2014 Published April 2015

Abstract
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This work deals with numerical approximations of unbounded and discontinuous value functions associated to some stochastic control problems. We derive error estimates for monotone schemes based on a Semi-Lagrangian method (or more generally in the form of a Markov chain approximation). A motivation of this study consists in approximating chance-constrained reachability sets. The latters will be characterized as level sets of a discontinuous value function associated to an adequate stochastic control problem. A precise analysis of the level-set approach is carried out and some numerical simulations are given to illustrate the approach.

Keywords: stochastic reachability analysis., numerical approximation, unbounded and discontinuous value function, level-set approach, chance constraints, Error estimates for HJB equations.

Mathematics Subject Classification: Primary: 65M15, 93E20; Secondary: 49L2.

Citation: Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933

References:

[1]	A. Abate, S. Amin, M. Prandini, J. Lygeros and S. Sastry, Computational approaches to reachability analysis of stochastic hybrid systems,, Hybrid Systems, 4416 (2007), 4. doi: 10.1007/978-3-540-71493-4_4. Google Scholar
[2]	A. Abate, M. Prandini, J. Lygeros and S. Sastry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems,, Automatica, 44 (2008), 2724. doi: 10.1016/j.automatica.2008.03.027. Google Scholar
[3]	M. Althoff, O. Stursberg and M. Buss, Safety assessement of autonomous cars using verification techniques,, American Control Conference, (2007), 4154. doi: 10.1109/ACC.2007.4282809. Google Scholar
[4]	M. Althoff, O. Stursberg and M. Buss, Safety assessement for stochastic linear systems using enclosing hulls of probability density functions,, European Control Conference, (): 625. Google Scholar
[5]	S. Amin, A. Abate, M. Prandini, S. Sastry and J. Lygeros, Reachability analysis for controlled discrete time stochastic hybrid systems,, in Lecture Notes in Computer Science LNCS, 3927 (2006), 49. doi: 10.1007/11730637_7. Google Scholar
[6]	G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations,, ESAIM:M2AN, 36 (2002), 33. doi: 10.1051/m2an:2002002. Google Scholar
[7]	G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations,, SIAM J. Numer. Anal., 43 (2005), 540. doi: 10.1137/S003614290343815X. Google Scholar
[8]	G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations,, Mathematics of Computations, 76 (2007), 1861. doi: 10.1090/S0025-5718-07-02000-5. Google Scholar
[9]	I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Difference quadrature schemes for nonlinear degenerate parabolic integro-pde,, SIAM J. Numer. Anal., 48 (2010), 1110. doi: 10.1137/090761501. Google Scholar
[10]	I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-diffusion processes,, Applied mathematics and optimization, 62 (2010), 47. doi: 10.1007/s00245-009-9095-8. Google Scholar
[11]	O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control and Optimization. Doi: 10.1137/090762075, 48 (2010), 4292. doi: 10.1137/090762075. Google Scholar
[12]	O. Bokanowski, A. Picarelli and H. Zidani, Dynamic programming and error estimates for stochastic control problems with maximum cost,, Applied Math. and Optimization, 71 (2015), 125. doi: 10.1007/s00245-014-9255-3. Google Scholar
[13]	J. Bonnans, S. Maroso and H. Zidani, Error bounds for stochastic differential games: The adverse stopping case,, IMA, 26 (2006), 188. doi: 10.1093/imanum/dri034. Google Scholar
[14]	J. Bonnans, S. Maroso and H. Zidani, Error estimates for a stochastic impulse control problem,, Applied. Math. and Optimisation, 55 (2007), 327. doi: 10.1007/s00245-006-0865-2. Google Scholar
[15]	B. Bouchard, R. Elie and N. Touzi, Stochastic target problems with controlled loss,, SIAM, 48 (2008), 3123. doi: 10.1137/08073593X. Google Scholar
[16]	A. Briani, F. Camilli and H. Zidani, Approximation schemes for monotone systems of nonlinear second order partial differential equations: convergence result and error estimate,, Differential Equations and Applications, 4 (2012), 297. doi: 10.7153/dea-04-18. Google Scholar
[17]	L. Caffarelli and P. E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pde,, Comm. Pure Appl. Math., 61 (2008), 1. doi: 10.1002/cpa.20208. Google Scholar
[18]	F. Camilli and E. R. Jakobsen, A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations,, SIAM J. Numer. Anal., 47 (2009), 2407. doi: 10.1137/080723144. Google Scholar
[19]	F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes,, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97. Google Scholar
[20]	F. Da Lio and O. Ley, Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data,, Applied Math. and Optimization, 63 (2011), 309. Google Scholar
[21]	K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations,, Mathematics of Computations, 82 (2013), 1433. doi: 10.1090/S0025-5718-2012-02632-9. Google Scholar
[22]	F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations,, J. Funct. Anal., 259 (2010), 1577. doi: 10.1016/j.jfa.2010.05.002. Google Scholar
[23]	W. H. Fleming and M. H. Soner, Controlled Markov Processes and Viscosity Solutions, vol. 25 of Stochastic Modelling and Applied Probability,, 2nd edition, (2006). Google Scholar
[24]	H. Föllmer and P. Leukert, Quantile hedging,, SIAM, 3 (1999), 251. doi: 10.1007/s007800050062. Google Scholar
[25]	D. Goreac and O.-S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost,, J. Math. Anal. Appl., 380 (2011), 327. doi: 10.1016/j.jmaa.2011.02.039. Google Scholar
[26]	N. V. Krylov, Mean value theorems for stochastic integrals,, Ann. Probab., 29 (2001), 385. doi: 10.1214/aop/1008956335. Google Scholar
[27]	N. Krylov, On the rate of convergence of finite difference approximation for Bellman's equation,, St. Petersburg Math. J., 9 (1998), 639. Google Scholar
[28]	N. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients,, Probability Theory and Related Fields, 117 (2000), 1. doi: 10.1007/s004400050264. Google Scholar
[29]	N. Krylov, On the rate of convergence for finite-difference approximations for bellman equations with lipschitz coefficients,, Applied Mathematics and Optimization, 52 (2005), 365. doi: 10.1007/s00245-005-0832-3. Google Scholar
[30]	H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics,, Springer, (2001). doi: 10.1007/978-1-4613-0007-6. Google Scholar
[31]	I. Mitchell, A. Bayen and C. Tomlin, A time-dependent Hamiliton-Jacobi formulation of reachable sets for continuous dynamic games,, IEEE Transactions on automatic control, 50 (2005), 947. doi: 10.1109/TAC.2005.851439. Google Scholar
[32]	R. Munos and H. Zidani, Consistency of a simple multidimensional scheme for hjb equations,, C. R. Acad. Sci. Paris, 340* (2005), 499. doi: 10.1016/j.crma.2005.02.001. Google Scholar*
[33]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[34]	R. Rubinstein and D. Kroese, Simulation and the Monte Carlo Method,, Wiley, (2008). Google Scholar
[35]	J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Stochastic Modelling and Applied Probability, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

show all references

References:

[1]	A. Abate, S. Amin, M. Prandini, J. Lygeros and S. Sastry, Computational approaches to reachability analysis of stochastic hybrid systems,, Hybrid Systems, 4416 (2007), 4. doi: 10.1007/978-3-540-71493-4_4. Google Scholar
[2]	A. Abate, M. Prandini, J. Lygeros and S. Sastry, Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems,, Automatica, 44 (2008), 2724. doi: 10.1016/j.automatica.2008.03.027. Google Scholar
[3]	M. Althoff, O. Stursberg and M. Buss, Safety assessement of autonomous cars using verification techniques,, American Control Conference, (2007), 4154. doi: 10.1109/ACC.2007.4282809. Google Scholar
[4]	M. Althoff, O. Stursberg and M. Buss, Safety assessement for stochastic linear systems using enclosing hulls of probability density functions,, European Control Conference, (): 625. Google Scholar
[5]	S. Amin, A. Abate, M. Prandini, S. Sastry and J. Lygeros, Reachability analysis for controlled discrete time stochastic hybrid systems,, in Lecture Notes in Computer Science LNCS, 3927 (2006), 49. doi: 10.1007/11730637_7. Google Scholar
[6]	G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations,, ESAIM:M2AN, 36 (2002), 33. doi: 10.1051/m2an:2002002. Google Scholar
[7]	G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations,, SIAM J. Numer. Anal., 43 (2005), 540. doi: 10.1137/S003614290343815X. Google Scholar
[8]	G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations,, Mathematics of Computations, 76 (2007), 1861. doi: 10.1090/S0025-5718-07-02000-5. Google Scholar
[9]	I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Difference quadrature schemes for nonlinear degenerate parabolic integro-pde,, SIAM J. Numer. Anal., 48 (2010), 1110. doi: 10.1137/090761501. Google Scholar
[10]	I. H. Biswas, E. R. Jakobsen and K. H. Karlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-diffusion processes,, Applied mathematics and optimization, 62 (2010), 47. doi: 10.1007/s00245-009-9095-8. Google Scholar
[11]	O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption,, SIAM J. Control and Optimization. Doi: 10.1137/090762075, 48 (2010), 4292. doi: 10.1137/090762075. Google Scholar
[12]	O. Bokanowski, A. Picarelli and H. Zidani, Dynamic programming and error estimates for stochastic control problems with maximum cost,, Applied Math. and Optimization, 71 (2015), 125. doi: 10.1007/s00245-014-9255-3. Google Scholar
[13]	J. Bonnans, S. Maroso and H. Zidani, Error bounds for stochastic differential games: The adverse stopping case,, IMA, 26 (2006), 188. doi: 10.1093/imanum/dri034. Google Scholar
[14]	J. Bonnans, S. Maroso and H. Zidani, Error estimates for a stochastic impulse control problem,, Applied. Math. and Optimisation, 55 (2007), 327. doi: 10.1007/s00245-006-0865-2. Google Scholar
[15]	B. Bouchard, R. Elie and N. Touzi, Stochastic target problems with controlled loss,, SIAM, 48 (2008), 3123. doi: 10.1137/08073593X. Google Scholar
[16]	A. Briani, F. Camilli and H. Zidani, Approximation schemes for monotone systems of nonlinear second order partial differential equations: convergence result and error estimate,, Differential Equations and Applications, 4 (2012), 297. doi: 10.7153/dea-04-18. Google Scholar
[17]	L. Caffarelli and P. E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pde,, Comm. Pure Appl. Math., 61 (2008), 1. doi: 10.1002/cpa.20208. Google Scholar
[18]	F. Camilli and E. R. Jakobsen, A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations,, SIAM J. Numer. Anal., 47 (2009), 2407. doi: 10.1137/080723144. Google Scholar
[19]	F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes,, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97. Google Scholar
[20]	F. Da Lio and O. Ley, Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data,, Applied Math. and Optimization, 63 (2011), 309. Google Scholar
[21]	K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations,, Mathematics of Computations, 82 (2013), 1433. doi: 10.1090/S0025-5718-2012-02632-9. Google Scholar
[22]	F. Delarue and S. Menozzi, Density estimates for a random noise propagating through a chain of differential equations,, J. Funct. Anal., 259 (2010), 1577. doi: 10.1016/j.jfa.2010.05.002. Google Scholar
[23]	W. H. Fleming and M. H. Soner, Controlled Markov Processes and Viscosity Solutions, vol. 25 of Stochastic Modelling and Applied Probability,, 2nd edition, (2006). Google Scholar
[24]	H. Föllmer and P. Leukert, Quantile hedging,, SIAM, 3 (1999), 251. doi: 10.1007/s007800050062. Google Scholar
[25]	D. Goreac and O.-S. Serea, Mayer and optimal stopping stochastic control problems with discontinuous cost,, J. Math. Anal. Appl., 380 (2011), 327. doi: 10.1016/j.jmaa.2011.02.039. Google Scholar
[26]	N. V. Krylov, Mean value theorems for stochastic integrals,, Ann. Probab., 29 (2001), 385. doi: 10.1214/aop/1008956335. Google Scholar
[27]	N. Krylov, On the rate of convergence of finite difference approximation for Bellman's equation,, St. Petersburg Math. J., 9 (1998), 639. Google Scholar
[28]	N. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients,, Probability Theory and Related Fields, 117 (2000), 1. doi: 10.1007/s004400050264. Google Scholar
[29]	N. Krylov, On the rate of convergence for finite-difference approximations for bellman equations with lipschitz coefficients,, Applied Mathematics and Optimization, 52 (2005), 365. doi: 10.1007/s00245-005-0832-3. Google Scholar
[30]	H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics,, Springer, (2001). doi: 10.1007/978-1-4613-0007-6. Google Scholar
[31]	I. Mitchell, A. Bayen and C. Tomlin, A time-dependent Hamiliton-Jacobi formulation of reachable sets for continuous dynamic games,, IEEE Transactions on automatic control, 50 (2005), 947. doi: 10.1109/TAC.2005.851439. Google Scholar
[32]	R. Munos and H. Zidani, Consistency of a simple multidimensional scheme for hjb equations,, C. R. Acad. Sci. Paris, 340* (2005), 499. doi: 10.1016/j.crma.2005.02.001. Google Scholar*
[33]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[34]	R. Rubinstein and D. Kroese, Simulation and the Monte Carlo Method,, Wiley, (2008). Google Scholar
[35]	J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Stochastic Modelling and Applied Probability, (1999). doi: 10.1007/978-1-4612-1466-3. Google Scholar

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