# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989
##### References:
 [1] J.-P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser Boston, Inc., Boston, Basel, Berlin, 1990. [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications. Boston, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. [3] M. Bardi and P. Soravia, A comparison result for Hamilton-Jacobi equations and applications to some differential games lacking controllability, Funkcial. Ekvac., 37 (1994), 19-43. [4] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, (French) [Viscosity solutions of Hamilton-Jacobi equations], Mathématiques & Applications, no. 17, Paris, Springer-Verlag, 1994. [5] E. N. Barron, The Pontryagin maximum principle for minimax problems of optimal control, Nonlinear Anal., 15 (1990), 1155-1165. doi: 10.1016/0362-546X(90)90051-H. [6] E. N. Barron, Viscosity solutions and analysis in $L^{\infty}$, 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. [7] E. N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost, Nonlinear Anal., Theory Methods Appl., 13 (1989), 1067-1090. doi: 10.1016/0362-546X(89)90096-5. [8] P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Int. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. [9] P. Bettiol and F. Rampazzo, ($L^\infty$ + Bolza) control problems as dynamic differential games, Nonlinear Differ. Equ. Appl., 20 (2013), 895-918. doi: 10.1007/s00030-012-0186-x. [10] P. Bettiol, H. Frankowska and R. B. Vinter, $L^{\infty}$ estimates on trajectories confined to a closed subset, J. Differential Eq., 252 (2012), 1912-1933. doi: 10.1016/j.jde.2011.09.007. [11] P. Bettiol and R. B. Vinter, Trajectories satisfying a smooth state constraint: Improved estimates, IEEE TAC, 56 (2011), 1090-1096. doi: 10.1109/TAC.2010.2088670. [12] P. Bettiol and R. B. Vinter, Estimates on trajectories in a closed set with corners for (t,x) dependent data, Mathematical Control and Related Fields, 3 (2013), 245-267. doi: 10.3934/mcrf.2013.3.245. [13] P. Bettiol and R. B. Vinter, Refined estimates on trajectories of state constrained control problems,, Preprint., (). [14] S. C. Di Marco and R. L. V. González, Minimax optimal control problems. Numerical analysis of the finite horizon case, ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 23-54. doi: 10.1051/m2an:1999103. [15] S. C. Di Marco and R. L. V. González, On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost, International Journal of Mathematics and Mathematical Sciences Issue, (2003), 4517-4538. doi: 10.1155/S0161171203302108. [16] L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Un. Math.J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040. [17] I. J. Fialho and T. T. Georgiou, Worst case analysis of nonlinear systems, IEEE Trans. Autom. Control, 44 (1999), 1180-1196. doi: 10.1109/9.769372. [18] H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differ. Equ. Appl., 20 (2013), 361-383. doi: 10.1007/s00030-012-0183-0. [19] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Eq., 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [20] J. Lygeros, On reachability and minimum cost optimal control, Automatica, 40 (2004), 917-927. doi: 10.1016/j.automatica.2004.01.012. [21] F. Rampazzo, Differential games with unbounded versus bounded controls, SIAM J. Control Optim., 36 (1998), 814-839. doi: 10.1137/S0363012995294602. [22] F. Rampazzo, Continuity of the upper and lower value of slow growth differential games, J. Math. Anal. Appl., 213 (1997), 15-31. doi: 10.1006/jmaa.1997.5327. [23] O. Serea, Discontinuity differential games and control systems with supremum cost, J. Math. Anal. Appl., 270 (2002), 519-542. doi: 10.1016/S0022-247X(02)00087-2. [24] R. B. Vinter, Minimax optimal control, SIAM J. Control Optim., 44 (2005), 939-968. doi: 10.1137/S0363012902415244. [25] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.

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##### References:
 [1] J.-P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser Boston, Inc., Boston, Basel, Berlin, 1990. [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems and Control: Foundations and Applications. Boston, Birkhäuser, 1997. doi: 10.1007/978-0-8176-4755-1. [3] M. Bardi and P. Soravia, A comparison result for Hamilton-Jacobi equations and applications to some differential games lacking controllability, Funkcial. Ekvac., 37 (1994), 19-43. [4] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, (French) [Viscosity solutions of Hamilton-Jacobi equations], Mathématiques & Applications, no. 17, Paris, Springer-Verlag, 1994. [5] E. N. Barron, The Pontryagin maximum principle for minimax problems of optimal control, Nonlinear Anal., 15 (1990), 1155-1165. doi: 10.1016/0362-546X(90)90051-H. [6] E. N. Barron, Viscosity solutions and analysis in $L^{\infty}$, 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. [7] E. N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost, Nonlinear Anal., Theory Methods Appl., 13 (1989), 1067-1090. doi: 10.1016/0362-546X(89)90096-5. [8] P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Int. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. [9] P. Bettiol and F. Rampazzo, ($L^\infty$ + Bolza) control problems as dynamic differential games, Nonlinear Differ. Equ. Appl., 20 (2013), 895-918. doi: 10.1007/s00030-012-0186-x. [10] P. Bettiol, H. Frankowska and R. B. Vinter, $L^{\infty}$ estimates on trajectories confined to a closed subset, J. Differential Eq., 252 (2012), 1912-1933. doi: 10.1016/j.jde.2011.09.007. [11] P. Bettiol and R. B. Vinter, Trajectories satisfying a smooth state constraint: Improved estimates, IEEE TAC, 56 (2011), 1090-1096. doi: 10.1109/TAC.2010.2088670. [12] P. Bettiol and R. B. Vinter, Estimates on trajectories in a closed set with corners for (t,x) dependent data, Mathematical Control and Related Fields, 3 (2013), 245-267. doi: 10.3934/mcrf.2013.3.245. [13] P. Bettiol and R. B. Vinter, Refined estimates on trajectories of state constrained control problems,, Preprint., (). [14] S. C. Di Marco and R. L. V. González, Minimax optimal control problems. Numerical analysis of the finite horizon case, ESAIM: Mathematical Modelling and Numerical Analysis, 33 (1999), 23-54. doi: 10.1051/m2an:1999103. [15] S. C. Di Marco and R. L. V. González, On a system of Hamilton-Jacobi-Bellman inequalities associated to a minimax problem with additive final cost, International Journal of Mathematics and Mathematical Sciences Issue, (2003), 4517-4538. doi: 10.1155/S0161171203302108. [16] L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Un. Math.J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040. [17] I. J. Fialho and T. T. Georgiou, Worst case analysis of nonlinear systems, IEEE Trans. Autom. Control, 44 (1999), 1180-1196. doi: 10.1109/9.769372. [18] H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differ. Equ. Appl., 20 (2013), 361-383. doi: 10.1007/s00030-012-0183-0. [19] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Eq., 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [20] J. Lygeros, On reachability and minimum cost optimal control, Automatica, 40 (2004), 917-927. doi: 10.1016/j.automatica.2004.01.012. [21] F. Rampazzo, Differential games with unbounded versus bounded controls, SIAM J. Control Optim., 36 (1998), 814-839. doi: 10.1137/S0363012995294602. [22] F. Rampazzo, Continuity of the upper and lower value of slow growth differential games, J. Math. Anal. Appl., 213 (1997), 15-31. doi: 10.1006/jmaa.1997.5327. [23] O. Serea, Discontinuity differential games and control systems with supremum cost, J. Math. Anal. Appl., 270 (2002), 519-542. doi: 10.1016/S0022-247X(02)00087-2. [24] R. B. Vinter, Minimax optimal control, SIAM J. Control Optim., 44 (2005), 939-968. doi: 10.1137/S0363012902415244. [25] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York-London, 1972.
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