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 and 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.

[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. 

[3]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier, North Holland, New York, Oxford, 1981.

[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.

[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.

[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.

[7]

J. Diestel and J. J. Uhl. Jr. Vector Measures, Mathematical Surveys and Monographs, AMS, Providence, R. I., 1977.

[8]

N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.

[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.

[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.

[11]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967.

[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.

[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.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic press, New York-London, 1975.

[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. 

[3]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, Elsevier, North Holland, New York, Oxford, 1981.

[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.

[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.

[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.

[7]

J. Diestel and J. J. Uhl. Jr. Vector Measures, Mathematical Surveys and Monographs, AMS, Providence, R. I., 1977.

[8]

N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.

[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.

[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.

[11]

K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York and London, 1967.

[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.

[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.

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