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

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

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.
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
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

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

[1]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

[2]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[3]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[4]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[5]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[6]

İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021010

[7]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[8]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[9]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[10]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[11]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006

[12]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[13]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3211-3240. doi: 10.3934/dcds.2020403

[14]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058

[15]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014

[16]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137

[17]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[18]

Sergey E. Mikhailov, Carlos F. Portillo. Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1103-1133. doi: 10.3934/cpaa.2021009

[19]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398

[20]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]