Homotopy for Rational Riccati Equations Arising in Stochastic Control

We consider the numerical solution of the rational algebraic Riccati equations in Rn, arising from stochastic optimal control in continuousand discrete-time. Applying the homotopy method, we continue from the maximal stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lypunov 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, where the relevant matrix operators are sparse, our algorithms are of O(n) complexity and memory requirement. Compared with the alternative (modified) Newton’s methods, our algorithms are easy to implement and do not require any difficult initialization. Some illustrative numerical examples are provided.