Abstract-The Riccati differential equation (RDE) arises in several fields like optimal control, optimal filtering, H∞ control of linear timevarying systems, differential games, etc. In the literature there is a large variety of approaches to compute its solution. Particularly for stiff RDEs, matrix-valued versions of the standard multi-step methods for solving ordinary differential equations have given good results. In this paper we discuss a particular class of one-step methods. These are the linearimplicit Runge-Kutta methods or Rosenbrock methods. We show that they offer a practical alternative for solving stiff RDEs. They can be implemented with good stability properties and allow for a cheap step size control. The matrix valued version of the Rosenbrock methods for RDEs requires the solution of one Sylvester equation in each stage of the method. For the case in which the coefficient matrices of the Sylvester equation are dense, the Bartels-Stewart method can be efficiently applied for solving the equations. The computational cost (computing time and memory requirements) is smaller than for the multi-step methods.
The efficient numerical integration of large-scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications. Standard numerical methods applied to such problems require an unduly amount of computing time and memory, in general. Based on a dynamical lowrank approximation of the solution, a new splitting integrator is proposed for a quite general class of stiff matrix differential equations. This class comprises differential Lyapunov and differential Riccati equations that arise from spatial discretizations of partial differential equations. The proposed integrator handles stiffness in an efficient way, and it preserves the symmetry and positive semidefiniteness of solutions of differential Lyapunov equations. Numerical examples that illustrate the benefits of this new method are given. In particular, numerical results for the efficient simulation of the weather phenomenon El Niño are presented.
Abstract:The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.