### Abstract

We consider the numerical solution of the rational algebraic Riccati equations in ℝ^{n}, arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n^{3}) computational complexity and O(n^{2}) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.

Original language | English |
---|---|

Pages (from-to) | B103-B125 |

Journal | SIAM Journal on Scientific Computing |

Volume | 37 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

### Fingerprint

### Keywords

- Generalized Stein equation
- Generalized lyapunov equation
- Rational algebraic Riccati equation
- Stochastic algebraic Riccati equation
- Stochastic optional control

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*37*(1), B103-B125. https://doi.org/10.1137/140953204

**Homotopy for rational riccati equations arising in stochastic optimal control.** / Zhang, Liping; Fan, Hung-Yuan; Chu, Eric King Wah; Wei, Yimin.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 37, no. 1, pp. B103-B125. https://doi.org/10.1137/140953204

}

TY - JOUR

T1 - Homotopy for rational riccati equations arising in stochastic optimal control

AU - Zhang, Liping

AU - Fan, Hung-Yuan

AU - Chu, Eric King Wah

AU - Wei, Yimin

PY - 2015/1/1

Y1 - 2015/1/1

N2 - We consider the numerical solution of the rational algebraic Riccati equations in ℝn, arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n3) computational complexity and O(n2) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.

AB - We consider the numerical solution of the rational algebraic Riccati equations in ℝn, arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n3) computational complexity and O(n2) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided.

KW - Generalized Stein equation

KW - Generalized lyapunov equation

KW - Rational algebraic Riccati equation

KW - Stochastic algebraic Riccati equation

KW - Stochastic optional control

UR - http://www.scopus.com/inward/record.url?scp=84923912581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923912581&partnerID=8YFLogxK

U2 - 10.1137/140953204

DO - 10.1137/140953204

M3 - Article

AN - SCOPUS:84923912581

VL - 37

SP - B103-B125

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 0036-1445

IS - 1

ER -