Efficient algorithms for solving stiff matrix-valued riccati differential equations

Abstract

One of the most deeply studied nonlinear matrix differential equations arising in science and engineering, notably in optimal control and filtering, is the Riccati differential equation (RDE). In the time-varying case, a classical approach which has been widely used to compute the solution of the matrix equation of size say, $n{\times}n$, is to unroll the matrices into vectors and integrate the resulting system of $n\sp2$ vector differential equations directly. If the system of vectorized differential equations is stiff, the cost (computation time and storage requirements) of applying the popular backward differentiation formulas (BDF's) to the stiff equations will be very high for large $n$ because a linear system of algebraic equations of the size $n\sp2{\times}n\sp2$ must be solved at each time step. By using a matrix generalization of the BDF's, i.e., by solving the matrix-valued differential equations directly, the amount of work required to compute the solution per time step is only $O(n\sp3$) flops, whereas the classical approach requires $O(n\sp6)$ flops per time step. A systematic compilation of properties of RDE's collected from various journals is included in this dissertation. In addition to those that lead to numerical algorithms, interesting results in asymptotic and monotonic properties are also given. Furthermore, conditions for the existence of periodic Riccati solutions are included. The results collected reflect the present state of research in the theory of RDE's. A survey of numerical methods for solving RDE's is presented. Eleven numerical methods for RDE's are described and are discussed in terms of their generality, efficiency, and numerical stability. The organized review of the numerical methods, together with that of the mathematical properties, should shed some light on the development of better theory and algorithms in the future. Efficient algorithms and reliable mathematical software are developed for the numerical solution of both explicit and implicit systems of linear differential equations. Numerical experiments confirm that the new algorithms are more efficient than the commonly used ones if the size of the systems is sufficiently large (say, $n$ $\geq$ 40).