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2005 | 25 | 1 | 129-157
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Measure valued solutions for stochastic evolution equations on Hilbert space and their feedback control

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In this paper, we consider a class of semilinear stochastic evolution equations on Hilbert space driven by a stochastic vector measure. The nonlinear terms are assumed to be merely continuous and bounded on bounded sets. We prove the existence of measure valued solutions generalizing some earlier results of the author. As a corollary, an existence result of a measure solution for a forward Kolmogorov equation with unbounded operator valued coefficients is obtained. The main result is further extended to cover Borel measurable drift and diffusion which are assumed to be bounded on bounded sets. Also we consider control problems for these systems and present several results on the existence of optimal feedback controls.
  • School of Information Technology and Engineering and Department of Mathematics, University of Ottawa
  • [1] N.U. Ahmed, Measure Solutions for Semilinear Evolution Equations with Polynomial Growth and Their Optimal Controls, Discuss. Math. Differential Inclusions 17 (1997), 5-27.
  • [2] N.U. Ahmed, Measure Solutions for Semilinear Systems with Unbounded Nonlinearities, Nonlinear Analysis: Theory, Methods and Applications 35 (1999), 487-503.
  • [3] N.U. Ahmed, Relaxed Solutions for Stochastic Evolution Equations on Hilbert Space with Polynomial Nonlinearities, Publicationes Mathematicae, Debrecen 54 (1-2) (1999), 75-101.
  • [4] N.U. Ahmed, Measure Solutions for Evolution Equations with Discontinuous Vector Fields, Nonlinear Functional analysis and Applications (NFAA) 9 (3)(2004), 467-484.
  • [5] N.U. Ahmed, Measure Solutions for Impulsive Evolution Equations with Measurable Vector Fields, JMAA, (submitted).
  • [6] S. Cerrai, Elliptic and parabolic equations in Rⁿ with coefficients having polynomial growth, Preprints di Matematica, n.9, Scuola Normale Superiore, Pisa 1995.
  • [7] A. Chojnowska-Michalik, Stochastic Differential Equations in Hilbert Spaces, Probability Theory, Banach Center Publications, 5, PWN-Polish Scientific Publishers 5 (1979), 53-74.
  • [8] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications 44 Cambridge University Press 1992.
  • [9] J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys Monogr. 15 AMS, Providence, RI 1977.
  • [10] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, Interscience Publishers, Inc., New York 1958.
  • [11] H.O. Fattorini, A Remark on Existence of solutions of Infinite Dimensional Non-compact Optimal Control Problems, SIAM J. Control and Optim. 35 (4) (1997), 1422-1433.
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