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Optimal Control for Stochastic Delay Evolution Equations

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Abstract

In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

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Acknowledgments

The authors thank the referee and the editor for their time, valuable comments and suggestions. This work was partially supported by the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholars (Grant No. LR15A010001), the National Natural Science Foundation of China (11101140, 1301177), China Postdoctoral Science Foundation (2011M500721, 2012T50391), and the Natural Science Foundation of Zhejiang Province (Nos.Y6110775, Y6110789)

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Authors and Affiliations

  1. Department of Mathematical Sciences, Huzhou University, Zhejiang, 313000, China

    Qingxin Meng

  2. Department of Mathematics and Statistics, York University, Toronto, ON, M3J 1P3, Canada

    Yang Shen

  3. School of Risk and Actuarial Studies and CEPAR, UNSW Business School, University of New South Wales, Sydney, NSW, 2052, Australia

    Yang Shen

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  1. Qingxin Meng
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  2. Yang Shen
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Correspondence to Yang Shen.

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Meng, Q., Shen, Y. Optimal Control for Stochastic Delay Evolution Equations. Appl Math Optim 74, 53–89 (2016). https://doi.org/10.1007/s00245-015-9308-2

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