# American Institute of Mathematical Sciences

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

 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.
Citation: Monica Motta, Caterina Sartori. Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data. Discrete and 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. [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. [3] M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413. [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. [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. [6] M. Bardi, A boundary value problem for the minimum-time function, SIAM J. Control Optim., 27 (1989), 776-785. doi: 10.1137/0327041. [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. [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. [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. [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. [11] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7), 2 (1988), 641-656. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

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##### 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. [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. [3] M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413. [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. [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. [6] M. Bardi, A boundary value problem for the minimum-time function, SIAM J. Control Optim., 27 (1989), 776-785. doi: 10.1137/0327041. [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. [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. [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. [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. [11] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B (7), 2 (1988), 641-656. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
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