A Multi-level Technique for the Approximate Solution of Opertaor Lyapunov and Riccati Equations

We consider multi-grid, or more appropriately, multi-level techniques for the numerical solution of operator Lyapunov and algebraic Riccati equations. The Riccati equation, which is quadratic, plays an essential role in the solution of linear-quadratic optimal control problems. The linear Lyapunov equation is important in the stability theory for linear systems and its solution is the primary step in the Newton-Kleinman algorithm for the solution of algebraic Riccati equations. Both equations are operator equations when the underlying linear system is in nite dimensional. In this case, nite dimensional discretization is required. However, as the level of discretization increases, the convergence rate of the standard iterative techniques for solving high order matrix Lyapunov and Riccati equations decreases. To deal with this, multi-leveling is introduced into the iterative Newton-Kleinman method for solving the algebraic Riccati equation, and Smith's method for solving matrix Lyapunov equations. Theoretical results and analysis indicating why the technique yields a signi cant improvement in e ciency over existing non-multi-grid techniques are provided, and the results of numerical studies on a test problem involving the optimal linear quadratic control of a one dimensional heat equation are discussed. This author was supported in part by the Air Force O ce of Scienti c Research under grant AFOSR 91-0076. This author was supported in part by the Air Force O ce of Scienti c Research under grant AFOSR 90-0091, and in part by a grant from the University of Southern California Faculty Research and Innovation Fund.