A Multi - Level Technique for the Approximate Solution of Operator Lyapunov andAlgebraic Riccati

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 innnite 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 signiicant improvement in eeciency 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.