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September  2015, 35(9): 3989-4017. doi: 10.3934/dcds.2015.35.3989

State constrained $L^\infty$ optimal control problems interpreted as differential games

Piernicola Bettiol 1,

1. 

Laboratoire de Mathematiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, 29200 Brest

Received  April 2014 Revised  October 2014 Published  April 2015

We consider state constrained optimal control problems in which the cost to minimize comprises an $L^\infty$ functional, i.e. the maximum of a running cost along the trajectories. In absence of state constraints, a new approach has been suggested by a recent paper [9]. The main purpose of the present paper is to extend this approach and the related results to state constrained $L^\infty$ optimal control problems. More precisely, using the $(L^\infty, L^1)$-duality, the reference optimal control problem can be seen as a static differential game, in which an extra variable is introduced and plays the role of an opponent player who wants to maximize the cost. Under appropriate assumptions and employing suitable Filippov's type results, this static game turns out to be equivalent to the corresponding dynamic differential game, whose (upper) value function is the unique viscosity solution to a constrained boundary value problem, which involves a Hamilton-Jacobi equation with a continuous Hamiltonian.
Keywords: Minimax problems, differential games, viscosity solutions..
Mathematics Subject Classification: Primary: 49K35, 49N70, 49L2.
Citation: Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989
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show all references

References:
[1]

Birkhäuser Boston, Inc., Boston, Basel, Berlin, 1990.  Google Scholar

[2]

Systems and Control: Foundations and Applications. Boston, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

Funkcial. Ekvac., 37 (1994), 19-43.  Google Scholar

[4]

(French) [Viscosity solutions of Hamilton-Jacobi equations], Mathématiques & Applications, no. 17, Paris, Springer-Verlag, 1994.  Google Scholar

[5]

Nonlinear Anal., 15 (1990), 1155-1165. doi: 10.1016/0362-546X(90)90051-H.  Google Scholar

[6]

Nonlinear Analysis, Differential Equations and Control, (Montreal, QC, 1998) (F. Clarke and R. J. Stern, eds.), NATO Sci. Ser. C Math. Phys. Sci., vol. 528, Kluwer Academic Publishers, DorFdrecht, (1999), 1-60.  Google Scholar

[7]

Nonlinear Anal., Theory Methods Appl., 13 (1989), 1067-1090. doi: 10.1016/0362-546X(89)90096-5.  Google Scholar

[8]

Int. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9.  Google Scholar

[9]

Nonlinear Differ. Equ. Appl., 20 (2013), 895-918. doi: 10.1007/s00030-012-0186-x.  Google Scholar

[10]

J. Differential Eq., 252 (2012), 1912-1933. doi: 10.1016/j.jde.2011.09.007.  Google Scholar

[11]

IEEE TAC, 56 (2011), 1090-1096. doi: 10.1109/TAC.2010.2088670.  Google Scholar

[12]

Mathematical Control and Related Fields, 3 (2013), 245-267. doi: 10.3934/mcrf.2013.3.245.  Google Scholar

[13]

P. Bettiol and R. B. Vinter, Refined estimates on trajectories of state constrained control problems,, Preprint., ().   Google Scholar

[14]

ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 23-54. doi: 10.1051/m2an:1999103.  Google Scholar

[15]

International Journal of Mathematics and Mathematical Sciences Issue, (2003), 4517-4538. doi: 10.1155/S0161171203302108.  Google Scholar

[16]

Indiana Un. Math.J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[17]

IEEE Trans. Autom. Control, 44 (1999), 1180-1196. doi: 10.1109/9.769372.  Google Scholar

[18]

Nonlinear Differ. Equ. Appl., 20 (2013), 361-383. doi: 10.1007/s00030-012-0183-0.  Google Scholar

[19]

J. Differential Eq., 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711.  Google Scholar

[20]

Automatica, 40 (2004), 917-927. doi: 10.1016/j.automatica.2004.01.012.  Google Scholar

[21]

SIAM J. Control Optim., 36 (1998), 814-839. doi: 10.1137/S0363012995294602.  Google Scholar

[22]

J. Math. Anal. Appl., 213 (1997), 15-31. doi: 10.1006/jmaa.1997.5327.  Google Scholar

[23]

J. Math. Anal. Appl., 270 (2002), 519-542. doi: 10.1016/S0022-247X(02)00087-2.  Google Scholar

[24]

SIAM J. Control Optim., 44 (2005), 939-968. doi: 10.1137/S0363012902415244.  Google Scholar

[25]

Academic Press, New York-London, 1972.  Google Scholar

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