Abstract
In this paper we present an upwind finite difference method for the numerical approximation of viscosity solutions of a two dimensional Hamilton-Jacobi-Bellman (HJB) equation arising from a class of optimal feedback control problems. The method is based on an explicit finite difference scheme and it is shown that the method is stable under certain constraints on the step lengths of the discretization. Numerical results, performed to verify the usefulness of the method, show that the method gives accurate approximate solutions to both of the control and the value function.
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References
Clarke, F.H. (1993), Optimization and Nonsmooth Analysis, Wiley, New York.
Ekeland, I. (1974), On the variational principle, J. Mathematical Analysis and Applications, Vol. 47, pp. 324–353.
Li, X. and Yong, J. (1995), Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston-Basel-Berlin.
Lions, P.L. (1983), Optimal control of diffusion process and Hamilton-Jacobi-Bellman equations, part 2: viscosity solutions and uniqueness, Comm. Partial Differential Equations, Vol. 11, pp. 1229–127.
Maslov, V.P., On a new principle of superposition for optimization problems, Russian Maths. Surveys, Vol. 42: 3, pp. 43–84.
Ortega, J.M. (1990), Numerical Analysis: A Second Course, SIAM, Philadelphia.
Peyret, R. and Taylor, T.D. (1983), Computational Methods for Fluid Flow, Springer-Verlag, New York.
Yong, J. (1992), The Method of Dynamic Programming and Hamilton-Jacobi-Bellman Equations,Shanghai Science & Technology Publisher, Shanghai (in Chinese).
Zhou, X.Y. (1990), Maximum principle, dynamic programming and their connection in deterministic control, J. Optimization Theory and Applications, Vol. 65, pp. 363–373.
Zhou X.Y. (1993), Verification theorems within the framework of viscosity solutions, J. Mathematical Analysis and Applications, Vol. 177, pp. 208–225.
Zhou, X.Y. and Sethi, S. (1993), A sufficient condition for near optimal stochastic controls and its applications to manufacturing systems, J. Applied Mathematics and Optimization, Vol. 177, pp. 208–225.
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© 2000 Kluwer Academic Publishers
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Wang, S., Gao, F., Teo, K.L. (2000). Solving Hamilton-Jacobi-Bellman Equations by an Upwind Finite Difference Method. In: Yang, X., Mees, A.I., Fisher, M., Jennings, L. (eds) Progress in Optimization. Applied Optimization, vol 39. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0301-5_17
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DOI: https://doi.org/10.1007/978-1-4613-0301-5_17
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7986-7
Online ISBN: 978-1-4613-0301-5
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