Abstract
Grid approximation schemes for constructing value functions and optimal feedbacks inproblems of guaranteed control are proposed. Value functions in optimal control problemsare usually nondifferentiable and corresponding feedbacks have a discontinuous switchingcharacter. Constructions of generalized gradients for local (convex, concave, linear) hullsare adapted to finite difference operators which approximate value functions. Optimal feedbacksare synthesized by extremal shift in the direction of generalized gradients. Bothproblems of constructing the value function and control synthesis are solved simultaneouslyin the single grid scheme. The interpolation problem is analyzed for grid values of optimalfeedbacks. Questions of correlation between spatial and temporal meshes are examined.The significance of quasiconvex properties is clarified for linear dependence of space‐timegrids.
The proposed grid schemes for solving optimal guaranteed control problems can beapplied to models arising in mechanics, mathematical economics, differential and evolutionarygames.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-P. Aubin, A survey of viability theory, SIAM J. Control Optimization 28(1990)749-788.
M. Bardi and S. Osher, The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations, SIAM J. Math. Anal. 22(1991)344-351.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277(1983)1-42.
M.G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comput. 43(1984)1-19.
G.G. Garnysheva and A.I. Subbotin, Strategies of minimax aiming in the direction of quasigradients, Prikl. Math. Mekh. 58(1994)5-11.
N.N. Krasovskii and A.I. Subbotin, Positional Differential Games, Nauka, Moscow, 1974.
N.N. Krasovskii, Control of a Dynamical System, Nauka, Moscow, 1985.
N.N. Krasovskii and A.I. Subbotin, Game-Theoretical Control Problems, Springer, New York Berlin, 1988.
A.N. Krasovskii and N.N. Krasovskii, Control under Lack of Information, Birkhäuser, Boston, 1995.
S. Osher and C.-W. Shu, High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal. 28(1991)907-922.
R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
P.E. Souganidis, Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, J. Different. Equat. 59(1985)1-43.
A.I. Subbotin, Generalization of the basic equation of the differential games theory, Dokl. Akad. Nauk SSSR. 254(1980)293-297.
A.I. Subbotin, Minimax Inequalities and Hamilton-Jacobi Equations, Nauka, Moscow, 1991.
A.I. Subbotin, Generalized Solutions for First-Order PDE. The Dynamical Optimization Perspective, Systems and Control: Foundations and Applications, Birkhäuser, Boston/Basel/Berlin, 1995.
A.I. Subbotin and A.M. Tarasyev, Conjugate derivatives of the value function of a differential game, Soviet Math. Dokl. 32(1985)162-166.
A.M. Tarasyev, Approximation schemes for constructing minimax solutions of Hamilton-Jacobi equations, J. Appl. Maths. Mechs. 58(1994)207-221.
A.M. Tarasyev, A.A. Uspenskii and V.N. Ushakov, Approximation schemes and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations, Izv. Akad. Sci. Tekhn. Kibernet. 3(1994)173-185.
Rights and permissions
About this article
Cite this article
Tarasyev, A. Control synthesis in grid schemesfor Hamilton‐Jacobi equations. Annals of Operations Research 88, 337–359 (1999). https://doi.org/10.1023/A:1018998817584
Issue Date:
DOI: https://doi.org/10.1023/A:1018998817584