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Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaacs’ Equations

Lions, Pierre-Louis; Souganidis, Panagiotis E. (1984), Differential Games, Optimal Control and Directional Derivatives of Viscosity Solutions of Bellman’s and Isaacs’ Equations, SIAM Journal on Control and Optimization, 23, 4, p. 566-583. http://dx.doi.org/10.1137/0323036

Type
Article accepté pour publication ou publié
Date
1984
Journal name
SIAM Journal on Control and Optimization
Volume
23
Number
4
Publisher
SIAM
Pages
566-583
Publication identifier
http://dx.doi.org/10.1137/0323036
Metadata
Show full item record
Author(s)
Lions, Pierre-Louis
Souganidis, Panagiotis E.
Abstract (EN)
Recent work by the authors and others has demonstrated the connections between the dynamic programming approach to optimal control theory and to two-person, zero-sum differential games problems and the new notion of “Viscosity” solutions of Hamilton–Jacobi PDE’s introduced by M. G. Crandall and P.-L. Lions. In particular, it has been proved that the dynamic programming principle implies that the value function is the viscosity solution of the associated Hamilton–Jacobi–Bellman and Isaacs equations. In the present work, it is shown that viscosity super- and subsolutions of these equations must satisfy some inequalities called super- and subdynamic programming principle respectively. This is then used to prove the equivalence between the notion of viscosity solutions and the conditions, introduced by A. Subbotin, concerning the sign of certain generalized directional derivatives.
Subjects / Keywords
differential games; optimal control; Hamilton-Jacobi equations; directional derivatives; viscosity solutions

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