# American Institute of Mathematical Sciences

December  2018, 11(6): 1011-1029. doi: 10.3934/dcdss.2018059

## Weak solutions of stochastic reaction diffusion equations and their optimal control

 EECS, University of Ottawa, Ottawa, K1N6N5, Canada

* Corresponding author: N.U.Ahmed

Received  January 2017 Revised  March 2017 Published  June 2018

Fund Project: The author is supported by NSERC grant A7109

In this paper we consider a class of stochastic reaction diffusion equations with polynomial nonlinearities. We prove existence and uniqueness of weak solutions and their regularity properties. We introduce a suitable topology on the space of stochastic relaxed controls and prove continuous dependence of solutions on controls with respect to this topology and the norm topology on the natural space of solutions. Also we prove that the attainable set of measures induced by the weak solutions is weakly compact. Then we consider some optimal control problems, including the Bolza problem, and some target seeking problems in terms of the attainable sets in the space of measures and prove existence of optimal controls. In the concluding section we present briefly some extensions of the results presented here.

Citation: N. U. Ahmed. Weak solutions of stochastic reaction diffusion equations and their optimal control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1011-1029. doi: 10.3934/dcdss.2018059
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic press, New York-London, 1975. Google Scholar [2] N. U. Ahmed, Optimal Control of reaction diffusion equations with potential applications to biomedical systems, Journal of Abstract Differential Equations and Applications, 8 (2017), 48-70. Google Scholar [3] N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier, North Holland, New York, Oxford, 1981. Google Scholar [4] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 37, Longman Scientific and Technical, U. K., 1988. Google Scholar [5] N. U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman Research Notes in Mathematics Series, 184. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Google Scholar [6] P. L. Chow, Explosive solutions of stochastic reaction diffusion equations in mean $L_p$-norm, Journal of Differential equations, 250 (2011), 2567-2580. doi: 10.1016/j.jde.2010.11.008. Google Scholar [7] J. Diestel and J. J. Uhl. Jr. Vector Measures, Mathematical Surveys and Monographs, AMS, Providence, R. I., 1977. Google Scholar [8] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. Google Scholar [9] X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity 1, Communication in Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar [10] C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, The Annals of Applied Probability, 9 (1999), 1226-1259. doi: 10.1214/aoap/1029962871. Google Scholar [11] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967. Google Scholar [12] J. Smoller, Shock Waves and Reaction-Diffusion Equations, (2nd Ed.), Springer Science and Business Media, LLC, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [13] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer- Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic press, New York-London, 1975. Google Scholar [2] N. U. Ahmed, Optimal Control of reaction diffusion equations with potential applications to biomedical systems, Journal of Abstract Differential Equations and Applications, 8 (2017), 48-70. Google Scholar [3] N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier, North Holland, New York, Oxford, 1981. Google Scholar [4] N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 37, Longman Scientific and Technical, U. K., 1988. Google Scholar [5] N. U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman Research Notes in Mathematics Series, 184. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Google Scholar [6] P. L. Chow, Explosive solutions of stochastic reaction diffusion equations in mean $L_p$-norm, Journal of Differential equations, 250 (2011), 2567-2580. doi: 10.1016/j.jde.2010.11.008. Google Scholar [7] J. Diestel and J. J. Uhl. Jr. Vector Measures, Mathematical Surveys and Monographs, AMS, Providence, R. I., 1977. Google Scholar [8] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. Google Scholar [9] X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: Diffusion and spatial heterogeneity 1, Communication in Pure and Applied Mathematics, 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar [10] C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, The Annals of Applied Probability, 9 (1999), 1226-1259. doi: 10.1214/aoap/1029962871. Google Scholar [11] K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967. Google Scholar [12] J. Smoller, Shock Waves and Reaction-Diffusion Equations, (2nd Ed.), Springer Science and Business Media, LLC, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar [13] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer- Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1988. doi: 10.1007/978-1-4684-0313-8. Google Scholar
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