A note on optimality conditions for optimal exit time problems

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June 2015, 5(2): 291-303. doi: 10.3934/mcrf.2015.5.291

A note on optimality conditions for optimal exit time problems

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

Received March 2014 Revised July 2014 Published April 2015

Abstract
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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

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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

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