A relaxation theorem for partially observed stochastic control on Hilbert space
Treść / Zawartość
In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which states that the set of solutions corresponding to ordinary controls is weakly dense in the set of solutions corresponding to relaxed controls. This is presented in Theorem 5.3 after giving some existence results on optimal controls for the infinite dimensional Zakai equation used for its proof.
-  G. Da Prato and J. Zabczyk, Regular Densities of Invariant Measures in Hilbert Spaces, Journal of Functional Analysis 130 (1995), 427-449 .
-  V. Barbu and G. Da Prato, Hamilton Jacobi Equations in Hilbert Spaces, Pitman Res. Notes in Math. 86, Pitman, 1983.
-  G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimension, Encyclopedia of Mathematics and it's Applications series, 44, Cambridge University Press, 1992.
-  G. Da Prato, Parabolic Equations in Infinitely Many Variables, Preprint n. 140, Scuola Normale Superiore, Pisa, Sept. 1992.
-  Q. Zhu and N.U. Ahmed, Some results on parabolic equations in Banach space, Nonlinear Analysis, Theory and Methods 24 (9) (1995), 1305-1319.
-  N.U. Ahmed, Generalized solutions of HJB equations applied to stochastic control on Hilbert space, Nonlinear Analysis 54 (2003), 495-523.
-  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, Existence of optimal relaxed controls for a class of systems governed by differential inclusions on a Banach space, JOTA 50 (2) (1986), 213-237.
-  N.U. Ahmed, Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions, International Symposium on Optimal Control of infinite Dimensional Systems, Lect. Notes in Pure and Applied Math, Marcel Dekker, New York and Basel 160 (1994), 1-19.
-  E.J. Balder, A general denseness result for relaxed control theory, Bull. Austral. Math. Soc. 30 (1984), 463-475.
-  H.O. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite dimensional systems, Journal of Differential Equations 112 (1994), 131-153.
-  D. Gątarek and B. Gołdys, On solving stochastic evolution equations by the change of drift with applications to optimal control, Polish Academy of Sciences, (personal communication).
-  W.H. Fleming and E. Pardoux, Optimal control for partially observed diffusions, SIAM J. Contr. Optim. 20 (1982), 261-285.
-  N.J. Cutland and T. Lindström, Random relaxed controls and partially observed stochastic systems, Acta Applicandae Mathematicae 32 (1993), 157-182.
-  O. Hijab, Partially observed control of Markov processes IV, J. Func. Anal. 109 (1992), 215-256.
-  A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University press, 1992.
-  V.S. Borkar, Existence of optimal controls for partially observed diffusions, Stochastics 11 (1983), 103-141.
-  N.U. Ahmed and X. Xiang, Admissible relaxation in optimal control problems for infinite dimensional uncertain systems, J. Appl. Math. and Stoch. Anal. 5 (3) (1993), 227-236.
-  N.U. Ahmed and J. Zabczyk, Partially observed optimal controls for nonlinear infinite dimensional stochastic systems, Dynamic Systems and Applications 5 (1996), 521-538.
-  J. Diestel and J.J. Uhl, Jr., Vector Measures, Mathematical Surveys, 15, AMS, American Mathematical Society, Providence, Rhode Island, 1977.
-  N.U. Ahmed and K.L. Teo, Optimal Control of Distributed Parameter Systems, North Holland, New York Oxford, 1981.
-  S.H. Hou, On property (Q) and other semicontinuity properties of multifunctions, Pacific J. Math. 103 (1) (1992), 39-56.
-  N.U. Ahmed, Optimal relaxed controls for infinite dimensional stochastic systems of Zakai type, SIAM J. Control and Optimization 34 (5) (1996), 1592-1615.
-  N.U. Ahmed, M. Fuhrman and J. Zabczyk, On filtering equations in infinite dimensions, J. Func. Anal. 143 (1) (1997), 180-204.
-  N. Dunford and J.T. Schwartz, Linear Operators, Part 1, Interscience Publishers, Inc., New York, 1964.
-  K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967.