Abstract
This paper describes the formal dynamic programming derivation of certain nonlinear PDE relevant in control theory and explains some recent work regarding the solution of these problems. These PDE are either first order (deterministic control) or second order, elliptic (stochastic control), and are either a system (if it costs to switch the control) or else a single fully nonlinear equation (if switching is free).
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References
L. Arnold, Stochastic Differential Equations, J. Wiley and Sons, 1974.
J. Ball and L.C. Evans, Weak convergence theorems for non-linear PDE of first and second order, Bull. London Math. Soc. 251 (1982), 332–346.
R. Bellman, Dynamic Programming, Princeton U. Press, 1957.
I. Capuzzo Dolcetta and L.C. Evans, Optimal switching for ordinary differential equations, to appear.
M. G. Crandall, L.C. Evans, and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, to appear in Trans. A.M.S.
M. G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, to appear in Trans. A.M.S.
L. C. Evans, A convergence theorem for solutions of nonlinear second order elliptic equations, Ind. U. Math. J. 27 (1978), 875–887.
L. C. Evans, On solving certain nonlinear PDE by accretive operator methods, Israel J. Math. 36 (1980), 225–247.
L. C. Evans, Classical solutions of the Hamilton-Jacobi- Bellman equation for uniformly elliptic operators, to appear in Trans. A.M.S.
L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. A.M.S. 253 (1979), 365–389.
L. C. Evans and P.L. Lions, Resolution des equations de Hamilton-Jacobi-Bellman, C.R.A.S. (1980).
W. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer, 1975.
I. Gihman and A.V. Skorohod, Controlled Stochastic Processes, Springer, 1979.
N. V. Krylov, Controlled Diffusion Processes, Springer, 1980.
P. L. Lions, Resolution analytic des problemes de Bellman- Dirichlet, Acta. Math. 146 (1981), 151–166.
P. L. Lions, Control of diffusion processes in]R, Comm. Pure. Appl. Math. 34 (1981), 121–147.
P. L. Lions, Sur les equations de Monge-Ampere I, II, III, to appear.
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pitman, 1982.
P. L. Lions and J. L. Menaldi, Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations I, II, SIAM J. Control 20 (1982), 58–95.
N. S. Trudinger, Fully nonlinear, uniformly elliptic equations under natural structure assumptions, to appear.
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© 1983 D. Reidel Publishing Company
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Evans, L.C. (1983). Nonlinear Systems in Optimal Control Theory and Related Topics. In: Ball, J.M. (eds) Systems of Nonlinear Partial Differential Equations. NATO Science Series C: (closed), vol 111. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7189-9_6
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DOI: https://doi.org/10.1007/978-94-009-7189-9_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-009-7191-2
Online ISBN: 978-94-009-7189-9
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