Abstract
The aim of this paper is to offer a quick overview of some applications of the theory of viscosity solutions of Hamilton-Jacobi-Bellman equations connected to nonlinear optimal control problems.
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References
Pesch H. J.; Bulirsch, R.: The Maximum Principle, Bellman’s equation, and Caratheodory’s work. J. Optim. Theory Appl. 80, 1994.
Kruzkov, S. N.: The Cauchy problem in the large for certain nonlinear first order differential equations. Sov. Math. Dokl. 1, 1960.
Kruzkov, S. N.: Generalized solutions of the Hamilton-Jacobi equations of eikonal type I. Math. USSR Sbornik 27, 1975.
Cannarsa P.; Sinestrari, C.: Convexity properties of the minimum time function. Calc. Var. 3, 1995.
Sinestrari, C.: Semiconcavity of solutions of stationary Hamilton-Jacobi equations. Nonlinear Analysis 24, 1995.
Crandall, M. G.; Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277, 1983.
Bardi, M.; Capuzzo Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. To appear, Birkhäuser 1997.
Lions, P. L.: Optimal control of diffusion processes and Hamilton-Jacobi equations. Comm. Partial Differential Equations 8, 1983.
Fleming, W. H.; Soner, M. H.: Controlled Markov processes and viscosity solutions. Springer Verlag 1993.
Li X.; Yong, J.: Optimal control theory for infinite dimensional systems. Birkhäuser 1995.
Capuzzo Dolcetta, I.: On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim. 10, 1983.
Capuzzo Dolcetta, I.; Ishii, H.: Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim. 11, 1984.
Bardi, M.: Some applications of viscosity solutions to optimal control and differential games. In Viscosity Solutions and Applications, I. Capuzzo Dolcetta, P. L. Lions (eds.). To appear in Lecture Notes in Mathematics, Springer 1997.
Barron, E. N.; Jensen, R.: Semicontinuous viscosity solutions of Hamilton-Jacobi equations with convex hamiltonians. Comm. Partial Differential Equations 15, 1990.
Barles, G.: Discontinuous viscosity solution of first order Hamilton-Jacobi equations: a guided visit. Nonlinear Analysis 20, 1993.
Soner, M. H.: Optimal control problems with state-space constraints I–II. SIAM J. Control Optim. 24, 1986.
Capuzzo Dolcetta, I.; Lions, P. L.: Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318, 1990.
Gonzale, R.; Rofman, E.: On deterministic control problems: an approximation procedure for the optimal cost. SIAM J. Control Optim. 23, 1985.
Falcone, M.: A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim. 15, 1987.
Rouy, A.: Numerical approximation of viscosity solutions of Hamilton-Jacobi equations with Neumann type boundary conditions. Math. Models Methods Appl. Sci. 2, 1992.
Falcone, M.; Ferretti, R.: Discrete high-order schemes for viscosity solutions of Hamilton-Jacobi equations. Numer. Math. 67, 1994.
Bardi, M.; Bottacin, S.; Falcone, M.: Convergence of discrete schemes for discontinuous value functions of pursuit-evasion games. In G. J. Olsder, editor, New Trends in Dynamic Games and Applications. Birkhäuser 1995.
Arisawa, M.: Ergodic problem for the Hamilton-Jacobi-Bellman equation I and II. Cahiers du CEREMADE. 1995.
Barles, G.: Solutions de Viscosite des Equations de Hamilton-Jacobi Vol. 17. Mathematiques et Applications. Springer 1994.
Bardi, M.; Bagagiolo, F.; Capuzzo Dolcetta, I.: A viscosity solutions approach to some asymptotic problems in optimal control. In J. P. Zolesio, editor, Proceedings of the Conference “PDE’s Methods in Control, Shape Optimization and Stochastic Modelling”. M. Dekker 1996.
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Dolcetta, I.C. (1998). Hamilton-Jacobi-Bellman Equations and Optimal Control. In: Schmidt, W.H., Heier, K., Bittner, L., Bulirsch, R. (eds) Variational Calculus, Optimal Control and Applications. International Series of Numerical Mathematics, vol 124. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8802-8_13
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DOI: https://doi.org/10.1007/978-3-0348-8802-8_13
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