NUMERICAL SOLUTIONS FOR STOCHASTIC CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATIONS

We are concerned with efficient numerical methods for stochastic continuous-time algebraic Riccati equations (SCARE). Such equations frequently arise from the state-dependent Riccati equation approach which is perhaps the only systematic way today to study nonlinear control problems. Often, involved Riccati-type equations are of small scale, but have to be solved repeatedly in real time. A new inner-outer iterative method that combines the fixed-point strategy and the structure-preserving doubling algorithm (SDA) is proposed. It is proved that the method is monotonically convergent to the desired stabilizing solution. Previously, Newton's method has been called to solve SCARE, but it was mostly investigated from its theoretical aspects rather than numerical aspects in terms of robust and efficient numerical implementation. For that reason, we revisit Newton's method for SCARE, focusing on how to make Newton's method practical. Finally numerical experiments are conducted to validate the new method and robust implementations of Newton's method.