Article ID: | iaor201526150 |
Volume: | 69 |
Issue: | 2 |
Start Page Number: | 271 |
End Page Number: | 290 |
Publication Date: | Jun 2015 |
Journal: | Numerical Algorithms |
Authors: | Blanes Sergio |
Keywords: | differential equations |
We consider the numerical integration of linear‐quadratic optimal control problems. This problem requires the solution of a boundary value problem: a non‐autonomous matrix Riccati differential equation (RDE) with final conditions coupled with the state vector equation with initial conditions. The RDE has positive definite matrix solution and to numerically preserve this qualitative property we propose first to integrate this equation backward in time with a sufficiently accurate scheme. Then, this problem turns into an initial value problem, and we analyse splitting and Magnus integrators for the forward time integration which preserve the positive definite matrix solutions for the RDE. Duplicating the time as two new coordinates and using appropriate splitting methods, high order methods preserving the desired property can be obtained. The schemes make sequential computations and do not require the storrage of intermediate results, so the storage requirements are minimal. The proposed methods are also adapted for solving linear‐quadratic