American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability and controllability of a wave equation with dynamical boundary control
  • MCRF Home
  • This Issue
  • Next Article
    Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension
June  2015, 5(2): 291-303. doi: 10.3934/mcrf.2015.5.291

A note on optimality conditions for optimal exit time problems

Luong V. Nguyen 1,

1. 

Università di Padova, Dipartimento di Matematica, via Trieste 63, 35121 Padova, Italy

Received  March 2014 Revised  July 2014 Published  April 2015

In this note, we obtain some optimality conditions for optimal control problems with exit time similar to those obtained in [Cannarsa, Pignotti and Sinestrari, Discrete Contin. Dynam. Systems 6 (2000), 975 - 997] without requiring an assumption on the Hamiltonian.
Keywords: semiconcave function, exit time problems., maximum principle, Value functions, optimality conditions.
Mathematics Subject Classification: Primary: 49K15, 93C15; Secondary: 49J5.
Citation: Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291
References:
[1]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamiltonian-Jacobi-Bellman Equations,, Systems & Control: Foundations & Applications, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

P. Cannarsa, A. Marigonda and K. T. Nguyen, Optimality conditions and regularity results for time optimal control problems with differential inclusions,, J. Math. Anal. Appl., 427 (2015), 202.  doi: 10.1016/j.jmaa.2015.02.027.  Google Scholar

[3]

P. Cannarsa, C. Pignotti and C. Sinestrari, Semiconcavity for optimal control problems with exit time,, Discrete Contin. Dynam. Systems, 6 (2000), 975.  doi: 10.3934/dcds.2000.6.975.  Google Scholar

[4]

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

[5]

P. Cannarsa and C. Sinestrari, On a class of nonlinear time optimal control problems,, Discrete Contin. Dynam. Systems, 1 (1995), 285.  doi: 10.3934/dcds.1995.1.285.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[7]

H. Frankowska and L. V. Nguyen, Local regularity of the minimum time function,, J. Optim. Theory. Appl., 164 (2005), 68.  doi: 10.1007/s10957-014-0575-x.  Google Scholar

[8]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, John Wiley & Sons, (1967).   Google Scholar

[9]

C. Pignotti, Rectifiability results for singular and conjugate points of optimal exit time problems,, J. Math. Anal. Appl., 270 (2002), 681.  doi: 10.1016/S0022-247X(02)00110-5.  Google Scholar

show all references

References:
[1]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamiltonian-Jacobi-Bellman Equations,, Systems & Control: Foundations & Applications, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[2]

P. Cannarsa, A. Marigonda and K. T. Nguyen, Optimality conditions and regularity results for time optimal control problems with differential inclusions,, J. Math. Anal. Appl., 427 (2015), 202.  doi: 10.1016/j.jmaa.2015.02.027.  Google Scholar

[3]

P. Cannarsa, C. Pignotti and C. Sinestrari, Semiconcavity for optimal control problems with exit time,, Discrete Contin. Dynam. Systems, 6 (2000), 975.  doi: 10.3934/dcds.2000.6.975.  Google Scholar

[4]

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

[5]

P. Cannarsa and C. Sinestrari, On a class of nonlinear time optimal control problems,, Discrete Contin. Dynam. Systems, 1 (1995), 285.  doi: 10.3934/dcds.1995.1.285.  Google Scholar

[6]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004).   Google Scholar

[7]

H. Frankowska and L. V. Nguyen, Local regularity of the minimum time function,, J. Optim. Theory. Appl., 164 (2005), 68.  doi: 10.1007/s10957-014-0575-x.  Google Scholar

[8]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory,, John Wiley & Sons, (1967).   Google Scholar

[9]

C. Pignotti, Rectifiability results for singular and conjugate points of optimal exit time problems,, J. Math. Anal. Appl., 270 (2002), 681.  doi: 10.1016/S0022-247X(02)00110-5.  Google Scholar

[1]

Carlo Sinestrari. Semiconcavity of the value function for exit time problems with nonsmooth target. Communications on Pure & Applied Analysis, 2004, 3 (4) : 757-774. doi: 10.3934/cpaa.2004.3.757
[2]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021
[3]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195
[4]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499
[5]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001
[6]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145
[7]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019089
[8]

M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137
[9]

Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975
[10]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483
[11]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659
[12]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial & Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783
[13]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[14]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[15]

Tianxiao Wang. Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I. Mathematical Control & Related Fields, 2019, 9 (2) : 385-409. doi: 10.3934/mcrf.2019018
[16]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557
[17]

Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487
[18]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43
[19]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017
[20]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences