Inclusions, Control and Optimization 21 (2001) 97-126
Department of Mathematics
University of Ottawa
Ottawa, Canada, K1N6N5
In this paper, we consider optimal feedback control for stochastc infinite dimensional systems. We present some new results on the solution of associated HJB equations in infinite dimensional Hilbert spaces. In the process, we have also developed some new mathematical tools involving distributions on Hilbert spaces which may have many other interesting applications in other fields. We conclude with an application to optimal stationary feedback control.
Keywords: optimal control, stochastic systems, infinite dimension, HJB equation, stationary feedback control.
2000 Mathematics Subject Classification: 93E20, 49J27, 49K45.
|||F. Gozzi and E. Rouy, Regular Solutions of Second-Order Stationary Hamilton-Jacobi Equations, Journal of Differential Equations 130 (1996), 210-234.|
|||G. Da Prato, Perturbations of Ornstein-Uhlenbeck Transition Semigroups by a Subquadratic Potential, Communications in Applied Analysis 2 (3) (1998), 431-444.|
|||B. Goldys and B. Maslowski, Ergodic Control of Semilinear Stochastic Equations and Hamilton-Jacobi Equations, preprint, (1998).|
|||G. Da Prato and J. Zabczyk, Regular Densities of Invariant Measures in Hilbert Spaces, Journal of Functional Analysis 130 (1995), 427-449.|
|||G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note, 229, Cambridge University Press 1996.|
|||G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Encyclopedia of Mathematics and its Applications series, 44, Cambridge University Press 1992.|
|||S. Cerrai, A Hille-Yosida Theorem for Weakly Continuous Semigroups, Semigroup Forum 49 (1994), 349-367.|
|||N.U. Ahmed and J. Zabczyk, Nonlinear Filtering for Semilinear Stochastic Differential Equations on Hilbert Spaces, Preprint 522, Inst. Math. Polish Academy of Sciences, Warsaw, Poland.|
|||N.U. Ahmed, Relaxed Controls for Stochastic Boundary Value Problems in Infinite Dimension, Lect. Notes in Contr. and Inf. Sciences 149 (1990), 1-10.|
|||N.U. Ahmed, Optimal Relaxed Controls for Nonlinear Infinite Dimensional Stochastic Differential Inclusions, International Symposium on Optimal Control of infinite Dimensional Systems, (ed. N.H. Pavel), Lect. Notes in Pure and Applied Math, Marcel Dekker, New York and Basel 160 (1994), 1-19.|
|||N.U. Ahmed, Optimal Relaxed Controls for Infinite Dimensional Stochastic Systems of Zakai Type, SIAM J. Control and Optimization 34 (5) (1996).|
|||N.U. Ahmed, M. Fuhrman and J. Zabczyk, On Filtering Equations in Infinite Dimensions, Journal of Functional Analysis 143 (1), 1997.|
|||S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol 1, Theory, Kluwer Academic Publishers, Dordrecht, Boston, London 1997.|
Received 30 June 2000