# A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces

## Abstract

We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solution of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.

- Authors:

- University of Innsbruck, Department of Mathematics (Austria)
- American University of Sharjah, Department of Mathematics (United Arab Emirates)

- Publication Date:

- OSTI Identifier:
- 22617041

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 75; Journal Issue: 3; Other Information: Copyright (c) 2017 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:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COMPUTER CALCULATIONS; CONTROL SYSTEMS; HILBERT SPACE; MANY-DIMENSIONAL CALCULATIONS; MATHEMATICAL SOLUTIONS; PARTIAL DIFFERENTIAL EQUATIONS; RICCATI EQUATION; STOCHASTIC PROCESSES

### Citation Formats

```
Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs, Mena, Hermann, E-mail: hermann.mena@uibk.ac.at, and Tuffaha, Amjad, E-mail: atufaha@aus.edu.
```*A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces*. United States: N. p., 2017.
Web. doi:10.1007/S00245-016-9339-3.

```
Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs, Mena, Hermann, E-mail: hermann.mena@uibk.ac.at, & Tuffaha, Amjad, E-mail: atufaha@aus.edu.
```*A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces*. United States. doi:10.1007/S00245-016-9339-3.

```
Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs, Mena, Hermann, E-mail: hermann.mena@uibk.ac.at, and Tuffaha, Amjad, E-mail: atufaha@aus.edu. Thu .
"A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces". United States.
doi:10.1007/S00245-016-9339-3.
```

```
@article{osti_22617041,
```

title = {A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces},

author = {Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs and Mena, Hermann, E-mail: hermann.mena@uibk.ac.at and Tuffaha, Amjad, E-mail: atufaha@aus.edu},

abstractNote = {We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solution of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.},

doi = {10.1007/S00245-016-9339-3},

journal = {Applied Mathematics and Optimization},

number = 3,

volume = 75,

place = {United States},

year = {Thu Jun 15 00:00:00 EDT 2017},

month = {Thu Jun 15 00:00:00 EDT 2017}

}