# Optimal Control for Stochastic Delay Evolution Equations

## 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.

- Authors:

- Huzhou University, Department of Mathematical Sciences (China)
- York University, Department of Mathematics and Statistics (Canada)

- Publication Date:

- OSTI Identifier:
- 22617267

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 74; Journal Issue: 1; Other Information: Copyright (c) 2016 Springer Science+Business Media New York; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CAUCHY PROBLEM; DIRICHLET PROBLEM; EVOLUTION; MATHEMATICAL SOLUTIONS; OPTIMAL CONTROL; PARTIAL DIFFERENTIAL EQUATIONS; RANDOMNESS; STOCHASTIC PROCESSES

### Citation Formats

```
Meng, Qingxin, E-mail: mqx@hutc.zj.cn, and Shen, Yang, E-mail: skyshen87@gmail.com.
```*Optimal Control for Stochastic Delay Evolution Equations*. United States: N. p., 2016.
Web. doi:10.1007/S00245-015-9308-2.

```
Meng, Qingxin, E-mail: mqx@hutc.zj.cn, & Shen, Yang, E-mail: skyshen87@gmail.com.
```*Optimal Control for Stochastic Delay Evolution Equations*. United States. doi:10.1007/S00245-015-9308-2.

```
Meng, Qingxin, E-mail: mqx@hutc.zj.cn, and Shen, Yang, E-mail: skyshen87@gmail.com. Mon .
"Optimal Control for Stochastic Delay Evolution Equations". United States.
doi:10.1007/S00245-015-9308-2.
```

```
@article{osti_22617267,
```

title = {Optimal Control for Stochastic Delay Evolution Equations},

author = {Meng, Qingxin, E-mail: mqx@hutc.zj.cn and Shen, Yang, E-mail: skyshen87@gmail.com},

abstractNote = {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.},

doi = {10.1007/S00245-015-9308-2},

journal = {Applied Mathematics and Optimization},

number = 1,

volume = 74,

place = {United States},

year = {Mon Aug 15 00:00:00 EDT 2016},

month = {Mon Aug 15 00:00:00 EDT 2016}

}