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

N1 - Publisher Copyright:
© 2015 Society for Industrial and Applied Mathematics.

PY - 2015

Y1 - 2015

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

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U2 - 10.1137/140953204

DO - 10.1137/140953204

M3 - Article

AN - SCOPUS:84923912581

SN - 1064-8275

VL - 37

SP - B103-B125

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 1

ER -