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September  2015, 35(9): 4527-4552. doi: 10.3934/dcds.2015.35.4527

Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data

Monica Motta 1, and  Caterina Sartori 2,

1. 

Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63, 35121 Padova
2. 

Dipartimento di Matematica, Via Trieste, 63, 35121 Padova, Italy

Received  April 2014 Revised  September 2014 Published  April 2015

In this paper we give a representation formula for the limit of the finite horizon problem as the horizon becomes infinite, with a nonnegative Lagrangian and unbounded data. It is related to the limit of the discounted infinite horizon problem, as the discount factor goes to zero. We give sufficient conditions to characterize the limit function as unique nonnegative solution of the associated HJB equation. We also briefly discuss the ergodic problem.
Keywords: unbounded viscosity solutions., optimal control problems with unbounded data, Asymptotic behaviour, impulsive controls.
Mathematics Subject Classification: Primary: 35B40; Secondary: 49J15, 49N25, 49L2.
Citation: Monica Motta, Caterina Sartori. Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4527-4552. doi: 10.3934/dcds.2015.35.4527
References:
[1]

O. Alvarez and E. N. Barron, Ergodic control in $L^\infty$. Set-valued analysis in control theory, Set-Valued Anal., 8 (2000), 51-69. doi: 10.1023/A:1008766206921.  Google Scholar

[2]

M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Bellman equation. II., Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 1-24. doi: 10.1016/S0294-1449(99)80019-5.  Google Scholar

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J. P. Aubin and A. Cellina, Differential Inclusions. Set-valued maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

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A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Second edition, Communications and Control Engineering Series. Springer-Verlag, Berlin, 2005. doi: 10.1007/b139028.  Google Scholar

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M. Bardi, A boundary value problem for the minimum-time function, SIAM J. Control Optim., 27 (1989), 776-785. doi: 10.1137/0327041.  Google Scholar

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M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Ed. Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[8]

M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 491-510. doi: 10.1007/s000300050027.  Google Scholar

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G. Barles and J. M. Roquejoffre, Ergodic type problems and large time behavior of unbounded solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 31 (2006), 1209-1225. doi: 10.1080/03605300500361461.  Google Scholar

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A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

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A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7), 2 (1988), 641-656.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, J. of Calc. Var. Partial Differential Equations, 3 (1995), 273-298. doi: 10.1007/BF01189393.  Google Scholar

[13]

D. A. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

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F. Da Lio, On the Bellman equation for infinite horizon problems with unbounded cost functional, Appl. Math. Optim., 41 (2000), 171-197. doi: 10.1007/s002459911010.  Google Scholar

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W. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition, Stochastic modelling and applied probability, Mathematics, 25. Springer-Verlag, New York, 2006.  Google Scholar

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M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 271-298. doi: 10.1007/s00030-004-1058-9.  Google Scholar

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Y. Giga, Q. Liu and H. Mitake, Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians, J. Differential Equations, 252 (2012), 1263-1282. doi: 10.1016/j.jde.2011.10.010.  Google Scholar

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R. Goebel, Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 357 (2005), 2187-2203. doi: 10.1090/S0002-9947-05-03817-1.  Google Scholar

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B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[20]

M. Motta, Viscosity solutions of HJB equations with unbounded data and characteristic points, Appl. Math. Optim., 49 (2004), 1-26. doi: 10.1007/s00245-003-0777-3.  Google Scholar

[21]

M. Motta and F. Rampazzo, Asymptotic controllability and optimal control, J. Differential Equations, 254 (2013), 2744-2763. doi: 10.1016/j.jde.2013.01.006.  Google Scholar

[22]

M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control, Optimisation and Calculus of Variations, 20 (2014), 957-982. doi: 10.1051/cocv/2014003.  Google Scholar

[23]

M. Motta and C. Sartori, The value function of an asymptotic exit-time optimal control problem, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 21-44, arXiv:1312.7443v2. doi: 10.1007/s00030-014-0274-1.  Google Scholar

[24]

M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535. doi: 10.3934/dcds.2008.21.513.  Google Scholar

[25]

M. Quincampoix and J. Renault, On the existence of a limit value in some nonexpansive optimal control problems, SIAM J. Control Optim., 49 (2011), 2118-2132. doi: 10.1137/090756818.  Google Scholar

[26]

F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.  Google Scholar

[27]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

show all references

References:
[1]

O. Alvarez and E. N. Barron, Ergodic control in $L^\infty$. Set-valued analysis in control theory, Set-Valued Anal., 8 (2000), 51-69. doi: 10.1023/A:1008766206921.  Google Scholar

[2]

M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Bellman equation. II., Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 1-24. doi: 10.1016/S0294-1449(99)80019-5.  Google Scholar

[3]

M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413.  Google Scholar

[4]

J. P. Aubin and A. Cellina, Differential Inclusions. Set-valued maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[5]

A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, Second edition, Communications and Control Engineering Series. Springer-Verlag, Berlin, 2005. doi: 10.1007/b139028.  Google Scholar

[6]

M. Bardi, A boundary value problem for the minimum-time function, SIAM J. Control Optim., 27 (1989), 776-785. doi: 10.1137/0327041.  Google Scholar

[7]

M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Ed. Birkhäuser, Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[8]

M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 491-510. doi: 10.1007/s000300050027.  Google Scholar

[9]

G. Barles and J. M. Roquejoffre, Ergodic type problems and large time behavior of unbounded solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 31 (2006), 1209-1225. doi: 10.1080/03605300500361461.  Google Scholar

[10]

A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[11]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7), 2 (1988), 641-656.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, J. of Calc. Var. Partial Differential Equations, 3 (1995), 273-298. doi: 10.1007/BF01189393.  Google Scholar

[13]

D. A. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control: Deterministic and Stochastic Systems, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

[14]

F. Da Lio, On the Bellman equation for infinite horizon problems with unbounded cost functional, Appl. Math. Optim., 41 (2000), 171-197. doi: 10.1007/s002459911010.  Google Scholar

[15]

W. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Second edition, Stochastic modelling and applied probability, Mathematics, 25. Springer-Verlag, New York, 2006.  Google Scholar

[16]

M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 271-298. doi: 10.1007/s00030-004-1058-9.  Google Scholar

[17]

Y. Giga, Q. Liu and H. Mitake, Large-time asymptotics for one-dimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians, J. Differential Equations, 252 (2012), 1263-1282. doi: 10.1016/j.jde.2011.10.010.  Google Scholar

[18]

R. Goebel, Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 357 (2005), 2187-2203. doi: 10.1090/S0002-9947-05-03817-1.  Google Scholar

[19]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.  Google Scholar

[20]

M. Motta, Viscosity solutions of HJB equations with unbounded data and characteristic points, Appl. Math. Optim., 49 (2004), 1-26. doi: 10.1007/s00245-003-0777-3.  Google Scholar

[21]

M. Motta and F. Rampazzo, Asymptotic controllability and optimal control, J. Differential Equations, 254 (2013), 2744-2763. doi: 10.1016/j.jde.2013.01.006.  Google Scholar

[22]

M. Motta and C. Sartori, On asymptotic exit-time control problems lacking coercivity, ESAIM Control, Optimisation and Calculus of Variations, 20 (2014), 957-982. doi: 10.1051/cocv/2014003.  Google Scholar

[23]

M. Motta and C. Sartori, The value function of an asymptotic exit-time optimal control problem, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 21-44, arXiv:1312.7443v2. doi: 10.1007/s00030-014-0274-1.  Google Scholar

[24]

M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535. doi: 10.3934/dcds.2008.21.513.  Google Scholar

[25]

M. Quincampoix and J. Renault, On the existence of a limit value in some nonexpansive optimal control problems, SIAM J. Control Optim., 49 (2011), 2118-2132. doi: 10.1137/090756818.  Google Scholar

[26]

F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.  Google Scholar

[27]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

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