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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 |
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. |
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. |
[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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