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Some applications of viscosity solutions to optimal control and differential games

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Viscosity Solutions and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1660))

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References

  • [A] B. Alziary de Roquefort, Jeux différentiels et approximation numérique de fonctions valeur, RAIRO Modél. Math. Anal. Numér. 25, pp. 517–560, 1991.

    MathSciNet  MATH  Google Scholar 

  • [BBCD] F. Bagagiolo, M. Bardi, I. Capuzzo Dolcetta A viscosity solutions approach to some asymptotic problems in optimal control, in Proceedings of the Conference “PDE’s methods in control, shape optimization and stochastic modelling”, Pisa 1994, J.P. Zolesio ed., Marcel Dekker, to appear.

    Google Scholar 

  • [B] M. Bardi, A boundary value problem for the minimum time function, SIAM J. Control Optim. 26, pp. 776–785, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [BB] M. Bardi, S. Bottacin, Discontinuous solutions of degenerate elliptic boundary value problems, preprint 22, Dipartimento di Matematica, Università di Padova, 1995.

    Google Scholar 

  • [BBF] M. Bardi, S. Bottacin, M. Falcone, Convergence of discrete schemes for discontinuous value functions of pursuit-evasion game, in “New Trends in Dynamic Games and Application”, G.J. Olsder ed., pp. 273–304. Birkhäuser, Boston, 1995.

    Chapter  Google Scholar 

  • [BCD] M. Bardi, I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Birkhäuser, Boston, to appear.

    Google Scholar 

  • [BF1] M. Bardi, M. Falcone, An approximation scheme for the minimum time function, SIAM J. Control Optim. 28, pp. 950–965, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [BF2] M. Bardi, M. Falcone, Discrete approximation of the minimal time function for systems with regular optimal trajectories, in “Analysis and Optimization of Systems”, A. Bensoussan and J.L. Lions eds., pp. 103–112, Lect. Notes Control Info. Sci. 144, Springer-Verlag, 1990.

    Google Scholar 

  • [BFS] M. Bardi, M. Falcone, P. Soravia, Fully discrete schemes for the value function of pursuit-evasion games, in “Advances in dynamic games and applications”, T. Basar and A. Haurie eds., pp. 89–105, Birkhäuser, 1993.

    Google Scholar 

  • [BPR] M. Bardi, T. Parthasarathy, T.E.S. Raghavan eds. Stochastic and differential games: theory and numerical methods, Birkhäuser, Boston, to appear.

    Google Scholar 

  • [BSa1] M. Bardi, C. Sartori, Approximations and regular perturbations of optimal control problems via Hamilton-Jacobi theory, Appl. Math. Optim. 24, pp. 113–128, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • [BSa2] M. Bardi, C. Sartori, Convergence results for Hamilton-Jacobi-Bellman equations in variable domains, Differential Integral Equations 5, pp. 805–816, 1992.

    MathSciNet  MATH  Google Scholar 

  • [BS0] M. Bardi, P. Soravia, A PDE framework for differential games of pursuitevasion type, in “Differential games and applications”, T. Basar and P. Bernhard eds., pp. 62–71, Lect. Notes Control Info. Sci. 119, Springer-Verlag, 1989.

    Google Scholar 

  • [BS1] M. Bardi, P. Soravia, Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games, Trans. Amer. Math. Soc. 325, pp. 205–229, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • [BS2] M. Bardi, P. Soravia, A comparison result for Hamilton-Jacobi equations and applications to some differential games lacking controllability, Funkcial. Ekvac. 37, pp. 19–43, 1994.

    MathSciNet  MATH  Google Scholar 

  • [BS3] M. Bardi, P. Soravia, Approximation of differential games of pursuit-evasion by discrete-time games, in “Differential Games—Developments in modelling and computation”, R.P. Hamalainen and H.K. Ethamo eds., pp. 131–143, Lect. Notes Control Info. Sci. 156, Springer-Verlag, 1991.

    Google Scholar 

  • [BSt] M. Bardi, V. Staicu, The Bellman equation for time-optimal control of noncontrollable nonlinear systems, Acta Applic. Math. 31, pp. 201–223, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ba0] G. Barles, An approach of deterministic control problems with unbounded data, Ann. Inst. H. Poincaré Anal. Nonlin. 7, pp.235–258, 1990.

    MathSciNet  MATH  Google Scholar 

  • [Ba1] G. Barles, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations: A guided visit, Nonlinear Anal. T.M.A. 20, pp. 1123–1134, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ba2] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag, Paris, 1994.

    MATH  Google Scholar 

  • [BP1] G. Barles, B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér. 21, pp. 557–579, 1987.

    MathSciNet  MATH  Google Scholar 

  • [BP2] G. Barles, B. Perthame, Exit time problems in optimal control and vanishing viscosity method, SIAM J. Control Optim. 26, pp. 1133–1148, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  • [BSou] G. Barles, P.E. Souganidis, Convergence of approximation schemes for fully nonlinear systems, Asymptotic Anal. 4, pp. 271–283, 1991.

    MathSciNet  MATH  Google Scholar 

  • [Br] E. N. Barron, Differential games with maximum cost, Nonlinear Anal. T. M. A. 14, pp. 971–989, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [BI] E.N. Barron, H. Ishii, The Bellman equation for minimizing the maximum cost, Nonlinear Anal. T.M.A. 13, pp. 1067–1090, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [BJ1] E.N. Barron, R. Jensen, Total risk aversion, stochastic optimal control, and differential games, Appl. Math. Optim. 19, pp. 313–327, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [BJ2] E.N. Barron, R. Jensen, Semicontinuous viscosity solutions of Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations, 15, pp. 1713–1742, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [BJM] E.N. Barron, R. Jensen, J.L. Menaldi, Optimal control and differential games with measures, Nonlinear Anal. T.M.A. 21, pp. 241–268, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [BaBe] T. Basar, P. Bernhard, H ∞-optimal control and related minimax design problems, Birkhäuser, Boston, 1991.

    MATH  Google Scholar 

  • [Be1] L.D. Berkovitz, Optimal feedback controls, SIAM J. Control Optim. 27, pp. 991–1006, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [Be2] L.D. Berkovitz, A theory of differential games, in “Advances in dynamic games and applications”, T. Basar and A. Haurie eds., pp. 3–22, Birkhäuser, 1993.

    Google Scholar 

  • [BiS] R.M. Bianchini, G. Stefani, Time-optimal problem and time-optimal map, Rend. Sem. Mat. Univ. Pol. Torino 48, pp. 401–429, 1990.

    MathSciNet  MATH  Google Scholar 

  • [Bo] S. Bortoletto, The Bellman equation for constrained deterministic optimal control problems, Differential Integral Equations 6, pp. 905–924, 1993.

    MathSciNet  MATH  Google Scholar 

  • [CFa1] F. Camilli, M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér. 29, pp. 97–122, 1995.

    MathSciNet  MATH  Google Scholar 

  • [CFa2] F. Camilli, M. Falcone, Approximation of optimal control problems with state constraints: estimates and applications, in “Nonsmooth analysis and geometric methods in deterministic optimal control”, V. Jurdjevic, B.S. Mordukhovic, R.T. Rockafellar and H.J. Sussman eds., I.M.A. Volumes in Applied Mathematics, Springer-Verlag, to appear.

    Google Scholar 

  • [CFLS] F. Camilli, M. Falcone, P. Lanucara, A. Seghini, A domain decomposition method for Bellman equations, in “Domain Decomposition methods in Scientific and Engineering Computing”, D.E. Keyes and J. Xu eds., pp. 477–483, Contemporary Mathematics 180, A.M.S., 1994.

    Google Scholar 

  • [CDP] P. Cannarsa, G. Da Prato, Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations, SIAM J. Control Optim. 27, pp. 861–875, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [CF] P. Cannarsa, H. Frankowska, Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim. 29, pp. 1322–1347, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • [CGS] P. Cannarsa, F. Gozzi, H. M. Soner, A dynamic programming approach to nonlinear boundary control problems of parabolic type, J. Funct. Anal. 117, pp. 25–61, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [CS] P. Cannarsa, C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. 3, pp. 273–298, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • [CD] I. Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. Optim. 10, pp. 367–377, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  • [CDE] I. Capuzzo Dolcetta, L.C. Evans, Optimal switching for ordinary differential equations, SIAM J. Control Optim. 22, pp. 143–161, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [CDF] I. Capuzzo Dolcetta, M. Falcone, Viscosity solutions and discrete dynamic programming, Ann. Inst. H. Poincaré Anal. Non Lin. 6 (Supplement), pp. 161–183, 1989.

    MathSciNet  MATH  Google Scholar 

  • [CDI] I. Capuzzo Dolcetta, H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. Optim. 11, pp. 161–181, 1984.

    Article  MathSciNet  Google Scholar 

  • [CDL] I. Capuzzo Dolcetta, P.L. Lions, Hamilton-Jacobi equations with state constraints, Trans. Amer. Math. Soc. 318, pp. 643–683, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ca] C. Caratheodory, Calculus of variations and partial differential equations of the first order, 2nd English edn., Chelsea, New York, 1982.

    MATH  Google Scholar 

  • [CQS1] P. Cardaliaguet, M. Quincampoix, P. Saint-Pierre, Temps optimaux pour des problèmes de controle avec contraintes et sans controlabilité locale, C. R. Acad. Sci. Paris 318, Ser. I, pp. 607–612, 1994.

    MathSciNet  Google Scholar 

  • [CQS2] P. Cardaliaguet, M. Quincampoix, P. Saint-Pierre, Set-valued numerical analysis for optimal control and differential games, in [BPR].

    Google Scholar 

  • [Cl1] F.H. Clarke, Optimization and nonsmooth analysis, Wiley, New York, 1983.

    MATH  Google Scholar 

  • [Cl2] F.H. Clarke, Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Reg. Conf. Ser. Appl. Math. 57, S.I.A.M., Philadelphia, 1989.

    Book  MATH  Google Scholar 

  • [C] M.G. Crandall, Viscosity Solutions: a Primer, in “Viscosity solutions and applications”, I. Capuzzo Dolcetta and P.L. Lions eds., Lecture Notes in Mathematics, Springer-Verlag, 1997.

    Google Scholar 

  • [CEL] M.G. Crandall, L.C. Evans, P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282, pp. 487–502, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [CIL] M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27, pp. 1–67, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • [CL1] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277, pp. 1–42, 1983.

    Article  MathSciNet  MATH  Google Scholar 

  • [CL2] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions, Part VI, in “Evolution equations, control theory, and biomathematics”, P. Clément and G. Lumer eds., Lect. Notes Pure Appl. Math. 155, Marcel Dekker, New York, 1994.

    Google Scholar 

  • [CL3] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions, Part VII, J. Func. Anal. 125, pp. 111–148, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • [D] F. Da Lio, Equazioni di Bellman per problemi di controllo ottimo illimitato, Thesis, Università di Padova, July 1994.

    Google Scholar 

  • [E] L. C. Evans, Nonlinear systems in optimal control theory and related topics, in “Systems of nonlinear PDEs”, J.M. Ball ed., pp. 95–113, D. Reidel, Dordrecht, 1983.

    Google Scholar 

  • [EI] L.C. Evans, H. Ishii, Differential games and nonlinear first order PDE in bounded domains, Manuscripta Math. 49, pp. 109–139, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [EJ] L.C. Evans, M.R. James, The Hamilton-Jacobi-Bellman equation for time optimal control, SIAM J. Control Optim. 27, pp. 1477–1489, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [ES] L.C. Evans, P.E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi equations, Indiana Univ. Math. J. 33, pp. 773–797, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [Fa] M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15, pp. 1–13, 1987 (Corrigenda in vol. 23, pp. 213–214, 1991).

    Article  MathSciNet  MATH  Google Scholar 

  • [FaF] M. Falcone, R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations, Numerische Mathematik 67, pp. 315–344, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • [F] W.H. Fleming, The convergence problem for differential games, J. Math. Anal. Appl. 3, pp. 102–116, 1961.

    Article  MathSciNet  MATH  Google Scholar 

  • [FME] W.H. Fleming, W.M. McEneaney, Risk sensitive optimal control and differential games, in Lecture Notes on Control and Information Sciences 184, pp. 185–197, Springer, 1992.

    Google Scholar 

  • [FS] W.H. Fleming, H.M. Soner, Controlled Markov processes and viscosity solutions, Springer, New York, 1993.

    MATH  Google Scholar 

  • [FSou] W.H. Fleming, P.E. Souganidis, On the existence of value function of two-players, zero-sum stochastic differential games, Indiana Univ. Math. J. 38, pp. 293–314, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [Fr] H. Frankowska, Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation, Appl. Math. Optim. 19, pp. 291–311, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  • [Fr2] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 31, pp. 257–272, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [G] P. Goatin, Sul problema di Dirichlet con condizioni al bordo generalizzate per equazioni ellittiche degeneri nonlineari, Thesis, Università di Padova, November 1995.

    Google Scholar 

  • [GT] R.L.V. Gonzalez, M.M. Tidball, Sur l’ordre de convergence des solutions discrétisées en temps et en espace de l’équation de Hamilton-Jacobi, C. R. Acad. Sci. Paris 314, Sér. I, pp. 479–482, 1992.

    MathSciNet  MATH  Google Scholar 

  • [Is] R. Isaacs, Differential games, Differential games, Wiley, New York, 1965.

    MATH  Google Scholar 

  • [I1] H. Ishii, Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, Indiana Univ. Math. J. 33, pp. 721–748 1984.

    Article  MathSciNet  MATH  Google Scholar 

  • [I2] H. Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55, pp. 369–384, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  • [I3] H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. Sc. Norm. Sup. Pisa (IV) 16, pp. 105–135, 1989.

    MathSciNet  MATH  Google Scholar 

  • [I4] H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces, J. Func. Anal. 105, pp. 301–341, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • [J] M.R. James, Asymptotic analysis of nonlinear stochastic risk-sensitive control and differential games, Math. Control Signals Systems 5, pp. 401–417, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ko] S. Koike, On the state constraint problem for differential games, Indiana Univ. Math. J. 44, pp. 467–487, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • [KK] A.N. Krasovskii, N.N. Krasovskii, Control under lack of information, Birkhäuser, Boston, 1995.

    Book  MATH  Google Scholar 

  • [KS] N.N. Krasovskii, A.I. Subbotin, Game theoretical control problems, Springer, New York, 1988.

    Book  Google Scholar 

  • [K] S.N. Kruzkov, Generalized solutions of the Hamilton-Jacobi equations of eikonal type I, Math. USSR Sbornik 27, pp. 406–445, 1975.

    Article  MATH  Google Scholar 

  • [KD] H.J. Kushner, P.G. Dupuis, Numerical methods for stochastic control problems in continuous time, Springer-Verlag, New York, 1992.

    Book  MATH  Google Scholar 

  • [Le] J. Lewin, Differential games, Springer, London, 1994.

    Book  Google Scholar 

  • [L1] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations Pitman, Boston, 1982.

    MATH  Google Scholar 

  • [L2] P.L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications, Part 2: Viscosity solutions and uniqueness, Comm. Partial. Differential Equations 8, (1983), 1101–1174 and 1229–1276.

    Article  MathSciNet  MATH  Google Scholar 

  • [L3] P.L. Lions, Neumann type boundary conditions for Hamilton-Jacobi equations, Duke J. Math. 52, pp. 793–820, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  • [L4] P.L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part III, J. Func. Anal. 86, pp. 1–18, 1989.

    Article  MATH  Google Scholar 

  • [LS] P.L. Lions, P.E. Souganidis, Differential games and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations I and II, SIAM J. Control Optim. 23 and 24, pp. 566–583 and 1086–1089, 1985 and 1986.

    Article  MathSciNet  MATH  Google Scholar 

  • [LT] P. Loreti, E. Tessitore, Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations, J. Math. Systems Estimation Control 4, pp. 467–483, 1994.

    MathSciNet  MATH  Google Scholar 

  • [ME] W.M. McEneaney, Uniqueness for viscosity solutions of nonstationary HJB equations under some a priori conditions (with applications), SIAM J. Control Optim. 33, pp. 1560–1576, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • [Mi1] S. Mirica, Inégalité différentielles impliquant l’équation de Bellman-Isaacs et ses généralisations dans la théorie du controle optimal, Anal. Univ. Bucuresti Mat. Anul 37, pp. 25–35, 1988.

    MathSciNet  MATH  Google Scholar 

  • [Mi2] S. Mirica, Some generalizations of the Bellman-Isaacs equation in deterministic optimal control, Studii Cercetari Mat. 42, pp. 437–447, 1990.

    MathSciNet  MATH  Google Scholar 

  • [Mi3] S. Mirica, Nonsmooth fields of extremals and constructive Dynamic Programming in optimal control, preprint 18, Dipartimento di Matematica, Università di Padova, 1993.

    Google Scholar 

  • [Mi4] S. Mirica, Optimal feedback control in closed form via Dynamic Programming, preprint 1993.

    Google Scholar 

  • [M] M. Motta, On nonlinear optimal control problems with state constraints, SIAM J. Control Optim. 33, pp. 1411–1424, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • [MR1] M. Motta, F. Rampazzo, Dynamic Programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim. to appear, 1996.

    Google Scholar 

  • [MR2] M. Motta, F. Rampazzo, The value function of a slow growth control problem with state constraints, J. Math. Systems Estimation Control to appear.

    Google Scholar 

  • [P1] H. J. Pesch, Offine and online computation of optimal trajectories in the aerospace field in “Applied Mathematics in Aerospace Science and Engeneering”, pp. 165–219, A. Miele and A. Salvetti eds., Plenum, New York, 1994.

    Chapter  Google Scholar 

  • [P2] H. J. Pesch, Solving optimal control and pursuit-evasion game problems of high complexity, in “Computational Optimal Control”, pp. 43–64, R. Bulirsch and D. Kraft, eds., Birkhäuser, Basel, 1994.

    Chapter  Google Scholar 

  • [R] F. Rampazzo, Differential games with unbounded versus bounded controls, preprint 30, Dipartimento di Matematica, Università di Padova, 1995.

    Google Scholar 

  • [RV] J. D. L. Rowland, R. B. Vinter, Construction of optimal feedback controls, Systems Control Lett. 16, pp. 357–367, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  • [S1] H.M. Soner, Optimal control problems with state-space constraints I and II, SIAM J. Control Optim. 24, pp. 551–561 and pp. 1110–1122, 1986.

    Google Scholar 

  • [S2] H.M. Soner, Controlled Markov Processes, Viscosity Solutions and Applications to Mathematical Finance, in “Viscosity solutions and applications”, I. Capuzzo Dolcetta and P.L. Lions eds., Lecture Notes in Mathematics, Springer-Verlag, 1997.

    Google Scholar 

  • [Sor0] P. Soravia, The concept of value in differential games of survival and viscosity solutions of Hamilton-Jacobi equations, Differential Integral Equations 5, pp. 1049–1068, 1992.

    MathSciNet  MATH  Google Scholar 

  • [Sor1] P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control Optim. 31, pp. 604–623, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sor2] P. Soravia, Discontinuous viscosity solutions to Dirichlet problems for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 18, pp. 1493–1514, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sor3] P. Soravia, H ∞ control of nonlinear systems: differential games and viscosity solutions, SIAM J. Control Optim., to appear.

    Google Scholar 

  • [Sor4] P. Soravia, Differential games and viscosity solutions to study the H ∞ control of nonlinear, partially observed systems, Preprint volume of the 6th International Symposium on Dynamic Games and Applications, M. Breton and G. Zaccour eds., Montreal, 1994.

    Google Scholar 

  • [Sor5] P. Soravia, Estimates of convergence of fully discrete schemes for the Isaacs equation of pursuit-evasion differential games via maximum principle, preprint, Dipartimento di Matematica, Università di Padova, 1995.

    Google Scholar 

  • [Sor6] P. Soravia, Stability of dynamical systems with competitive controls: the degenerate case, J. Math. Anal. Appl. 191, pp. 428–449, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sor7] P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations, I and II, preprints, Dipartimento di Matematica, Università di Padova, 1995.

    Google Scholar 

  • [Sor8] P. Soravia, Equivalence between nonlinear H ∞ control problems and existence of viscosity solutions of Hamilton-Jacobi-Isaacs equations, preprint, Dipartimento di Matematica, Università di Padova, 1995.

    Google Scholar 

  • [Sou] P.E. Souganidis, Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games, Nonlinear Anal. T.M.A. 9, pp. 217–257, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sul] A.I. Subbotin, Discontinuous solutions of a Dirichlet type boundary value problem for first order partial differential equations, Russian J. Numer. Anal. Math. Modelling 8, pp. 145–164, 1993.

    MathSciNet  MATH  Google Scholar 

  • [Su2] A.I. Subbotin, Generalized solutions of first order PDEs: The Dynamic Optimization Perspective, Birkhäuser, Boston, 1995.

    Book  Google Scholar 

  • [Su3] A.I. Subbotin, Constructive theory of positional differential games and generalized solutions to Hamilton-Jacobi equations, in [BPR].

    Google Scholar 

  • [Sua] N.N. Subbotina, The maximum principle and the superdifferential of the value function, Prob. Control Inform. Th. 18, pp. 151–160, 1989.

    MathSciNet  MATH  Google Scholar 

  • [T] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach, J. Differential Equations 111, pp. 123–146, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • [TG] M.M. Tidball and R.L.V. Gonzalez, Zero sum differential games with stopping times: some results about their numerical resolution, in “Advances in dynamic games and applications”, T. Basar and A. Haurie eds., pp. 106–124, Birkhäuser, 1993.

    Google Scholar 

  • [Y1] J. Yong, A zero-sum differential game in a finite duration with switching strategies, SIAM J. Control Optim. 28, pp. 1234–1250, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [Y2] J. Yong, Zero-sum differential games involving impulse controls, Appl. Math. Optim. 29, pp. 243–261, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  • [Z1] X.Y. Zhou, Maximum principle, dynamic programming, and their connection in deterministic control, J. Optim. Th. Appl. 65, pp. 363–373, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  • [Z2] X.Y. Zhou, Verification theorems within the framework of viscosity solutions, J. Math. Anal. Appl. 177, pp. 208–225, 1993.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni, 7, I-35131, Padova, Italy

    Martino Bardi

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  1. Martino Bardi
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Italo Capuzzo Dolcetta Pierre Louis Lions

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© 1997 Springer-Verlag

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Bardi, M. (1997). Some applications of viscosity solutions to optimal control and differential games. In: Dolcetta, I.C., Lions, P.L. (eds) Viscosity Solutions and Applications. Lecture Notes in Mathematics, vol 1660. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094295

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  • DOI: https://doi.org/10.1007/BFb0094295

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62910-8

  • Online ISBN: 978-3-540-69043-6

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