# 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 & Continuous Dynamical Systems - A, 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, (1990). Google Scholar [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Systems and Control: Foundations and Applications. Boston, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar [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. Google Scholar [4] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, (French) [Viscosity solutions of Hamilton-Jacobi equations], (1994). Google Scholar [5] E. N. Barron, The Pontryagin maximum principle for minimax problems of optimal control,, Nonlinear Anal., 15 (1990), 1155. doi: 10.1016/0362-546X(90)90051-H. Google Scholar [6] E. N. Barron, Viscosity solutions and analysis in $L^{\infty}$,, Nonlinear Analysis, (1999), 1. Google Scholar [7] E. N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost,, Nonlinear Anal., 13 (1989), 1067. doi: 10.1016/0362-546X(89)90096-5. Google Scholar [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. doi: 10.1007/s00182-006-0030-9. Google Scholar [9] P. Bettiol and F. Rampazzo, ($L^\infty$ + Bolza) control problems as dynamic differential games,, Nonlinear Differ. Equ. Appl., 20 (2013), 895. doi: 10.1007/s00030-012-0186-x. Google Scholar [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. doi: 10.1016/j.jde.2011.09.007. Google Scholar [11] P. Bettiol and R. B. Vinter, Trajectories satisfying a smooth state constraint: Improved estimates,, IEEE TAC, 56 (2011), 1090. doi: 10.1109/TAC.2010.2088670. Google Scholar [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. 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] 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. doi: 10.1051/m2an:1999103. Google Scholar [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. doi: 10.1155/S0161171203302108. Google Scholar [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. doi: 10.1512/iumj.1984.33.33040. Google Scholar [17] I. J. Fialho and T. T. Georgiou, Worst case analysis of nonlinear systems,, IEEE Trans. Autom. Control, 44 (1999), 1180. doi: 10.1109/9.769372. Google Scholar [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. doi: 10.1007/s00030-012-0183-0. Google Scholar [19] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains,, J. Differential Eq., 161 (2000), 449. doi: 10.1006/jdeq.2000.3711. Google Scholar [20] J. Lygeros, On reachability and minimum cost optimal control,, Automatica, 40 (2004), 917. doi: 10.1016/j.automatica.2004.01.012. Google Scholar [21] F. Rampazzo, Differential games with unbounded versus bounded controls,, SIAM J. Control Optim., 36 (1998), 814. doi: 10.1137/S0363012995294602. Google Scholar [22] F. Rampazzo, Continuity of the upper and lower value of slow growth differential games,, J. Math. Anal. Appl., 213 (1997), 15. doi: 10.1006/jmaa.1997.5327. Google Scholar [23] O. Serea, Discontinuity differential games and control systems with supremum cost,, J. Math. Anal. Appl., 270 (2002), 519. doi: 10.1016/S0022-247X(02)00087-2. Google Scholar [24] R. B. Vinter, Minimax optimal control,, SIAM J. Control Optim., 44 (2005), 939. doi: 10.1137/S0363012902415244. Google Scholar [25] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar

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##### References:
 [1] J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Birkhäuser Boston, (1990). Google Scholar [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,, Systems and Control: Foundations and Applications. Boston, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar [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. Google Scholar [4] G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi,, (French) [Viscosity solutions of Hamilton-Jacobi equations], (1994). Google Scholar [5] E. N. Barron, The Pontryagin maximum principle for minimax problems of optimal control,, Nonlinear Anal., 15 (1990), 1155. doi: 10.1016/0362-546X(90)90051-H. Google Scholar [6] E. N. Barron, Viscosity solutions and analysis in $L^{\infty}$,, Nonlinear Analysis, (1999), 1. Google Scholar [7] E. N. Barron and H. Ishii, The Bellman equation for minimizing the maximum cost,, Nonlinear Anal., 13 (1989), 1067. doi: 10.1016/0362-546X(89)90096-5. Google Scholar [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. doi: 10.1007/s00182-006-0030-9. Google Scholar [9] P. Bettiol and F. Rampazzo, ($L^\infty$ + Bolza) control problems as dynamic differential games,, Nonlinear Differ. Equ. Appl., 20 (2013), 895. doi: 10.1007/s00030-012-0186-x. Google Scholar [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. doi: 10.1016/j.jde.2011.09.007. Google Scholar [11] P. Bettiol and R. B. Vinter, Trajectories satisfying a smooth state constraint: Improved estimates,, IEEE TAC, 56 (2011), 1090. doi: 10.1109/TAC.2010.2088670. Google Scholar [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. 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] 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. doi: 10.1051/m2an:1999103. Google Scholar [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. doi: 10.1155/S0161171203302108. Google Scholar [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. doi: 10.1512/iumj.1984.33.33040. Google Scholar [17] I. J. Fialho and T. T. Georgiou, Worst case analysis of nonlinear systems,, IEEE Trans. Autom. Control, 44 (1999), 1180. doi: 10.1109/9.769372. Google Scholar [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. doi: 10.1007/s00030-012-0183-0. Google Scholar [19] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains,, J. Differential Eq., 161 (2000), 449. doi: 10.1006/jdeq.2000.3711. Google Scholar [20] J. Lygeros, On reachability and minimum cost optimal control,, Automatica, 40 (2004), 917. doi: 10.1016/j.automatica.2004.01.012. Google Scholar [21] F. Rampazzo, Differential games with unbounded versus bounded controls,, SIAM J. Control Optim., 36 (1998), 814. doi: 10.1137/S0363012995294602. Google Scholar [22] F. Rampazzo, Continuity of the upper and lower value of slow growth differential games,, J. Math. Anal. Appl., 213 (1997), 15. doi: 10.1006/jmaa.1997.5327. Google Scholar [23] O. Serea, Discontinuity differential games and control systems with supremum cost,, J. Math. Anal. Appl., 270 (2002), 519. doi: 10.1016/S0022-247X(02)00087-2. Google Scholar [24] R. B. Vinter, Minimax optimal control,, SIAM J. Control Optim., 44 (2005), 939. doi: 10.1137/S0363012902415244. Google Scholar [25] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar
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