Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T09:13:37.879Z Has data issue: false hasContentIssue false
  • English
  • Français

Averaging of the Hamilton-Jacobi equation in infinite dimensions and an application

Published online by Cambridge University Press:  17 February 2009

Shihong Wang  and
Zuoyi Zhou
Shihong Wang
Affiliation:
Institute of Mathematics, Fudan University, Shanghai, China
Zuoyi Zhou
Affiliation:
Institute of Mathematics, Fudan University, Shanghai, China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 3 , January 2000 , pp. 372 - 385
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Barron, E. N., “Averaging in Lagrange and minimax problems of optimal control”, SIAM J. Control and Optim. 31 (1993) 1630–1652.CrossRefGoogle Scholar
[2]Bensoussan, A., Lions, J. L. and Papanicolaon, G., Asymptotic Analysis for Periodic Structures (North Holland, New York, 1987).Google Scholar
[3]Chaplais, F., “Averaging and deterministic optimal control”, SIAM J. Control and Optim. 25 (1987) 767–780.CrossRefGoogle Scholar
[4]Crandall, M. G., Evans, L. C. and Lions, P. L., “Some properties of viscosity solutions of Hamilton-Jacobi equations”, Trans. Amer. Math. Soc. 282 (1984) 487–502.CrossRefGoogle Scholar
[5]Crandall, M. G. and Lions, P. L., “Viscosity solutions of Hamilton-Jacobi equations”, Trans. Amer. Math. Soc. 277 (1983) 1–42.CrossRefGoogle Scholar
[6]Crandall, M. G. and Lions, P. L., “Hamilton-Jacobi equations in infinite dimension. I Uniqueness of viscosity solutions”, J. Funct. Anal. 62 (1985) 379–396.CrossRefGoogle Scholar
[7]Crandall, M. G. and Lions, P. L., “Hamilton-Jacobi equations in infinite dimension. II Existence of viscosity solutions”, J. Funct. Anal. 65 (1986) 368–405.CrossRefGoogle Scholar
[8]Crandall, M. G. and Lions, P. L., “Hamilton-Jacobi equations in infinite dimension. IV Hamilton with unbounded linear term”, J. Funct. Anal. 90 (1990) 237–283.CrossRefGoogle Scholar
[9]Li, X. J. and Yong, J. M., Optimal Control Theory for Infinite Dimensional Systems (Birkhäuser, Boston, 1995).CrossRefGoogle Scholar
[10]Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations (Pitman, London, 1982).Google Scholar
[11]Pazy, A., Semigroup of Linear Operators and Application to Partial Differential Equations (Springer, New York, Berlin, 1983).CrossRefGoogle Scholar
[12]Stegall, C., “Optimization of functions on certain subsets of Banach space”, Math. Ann. 236 (1978) 171–176.CrossRefGoogle Scholar
[13]Yong, J. M., Dynamic Programming Method and HJB Equation (Shanghai Science Technology Press, Shanghai, 1992).Google Scholar