Modified Douglas splitting method for differential matrix equations

Abstract In this paper, we consider a modified Douglas splitting method for a class of differential matrix equations, including differential Lyapunov and differential Riccati equations. The method we consider is based on a natural three-term splitting of the equations. The implementation of the algorithm requires only the solution of a linear algebraic system with multiple right-hand sides in each time step. It is proved that the method is convergent of order two and it preserves the symmetry and positive semidefiniteness of solutions of differential Lyapunov equations. Moreover, we show how the method can be handled in a low-rank setting for large-scale computations. We also provide a theoretical a priori error analysis for the low-rank algorithms. Numerical results are presented to verify the theoretical analysis.

[1]  Alexander Ostermann,et al.  Convergence of a Low-Rank Lie-Trotter Splitting for Stiff Matrix Differential Equations , 2018, SIAM J. Numer. Anal..

[2]  Ninoslav Truhar,et al.  Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix , 2007, Syst. Control. Lett..

[3]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[4]  L. Dieci,et al.  Positive definiteness in the numerical solution of Riccati differential equations , 1994 .

[5]  Tony Stillfjord,et al.  Convergence Analysis for Splitting of the Abstract Differential Riccati Equation , 2014, SIAM J. Numer. Anal..

[6]  Zvonimir Bujanovic,et al.  RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations , 2015, Numerische Mathematik.

[7]  Tony Stillfjord,et al.  Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations , 2015, IEEE Transactions on Automatic Control.

[8]  Mena Pazmiño,et al.  Numerical solutions of differential Riccati Equations Arising in Optimal Control Problems for Parabolic Partial Differential Equations , 2007 .

[9]  Peter Benner,et al.  Rosenbrock Methods for Solving Riccati Differential Equations , 2013, IEEE Transactions on Automatic Control.

[10]  Akira Ichikawa,et al.  Remarks on the time-varying H∞ Riccati equations , 1999 .

[11]  Hermann Mena,et al.  An LDLT factorization based ADI algorithm for solving large‐scale differential matrix equations , 2014 .

[12]  I. Petersen,et al.  Robust Control Design Using H-? Methods , 2012 .

[13]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[14]  J. J. Douglas.  Alternating direction methods for three space variables , 1962 .

[15]  P. Benner,et al.  Peer Methods for the Solution of Large-Scale Differential Matrix Equations , 2018, 1807.08524.

[16]  Tony Stillfjord,et al.  Adaptive high-order splitting schemes for large-scale differential Riccati equations , 2016, Numerical Algorithms.

[17]  Valeria Simoncini,et al.  Computational Methods for Linear Matrix Equations , 2016, SIAM Rev..

[18]  Chiara Piazzola,et al.  Numerical low-rank approximation of matrix differential equations , 2017, J. Comput. Appl. Math..

[19]  Y. Zhou,et al.  On the decay rate of Hankel singular values and related issues , 2002, Syst. Control. Lett..

[20]  Hermann Mena,et al.  A Structure Preserving Krylov Subspace Method for Large Scale Differential Riccati Equations , 2017 .

[21]  J. Douglas,et al.  A general formulation of alternating direction methods , 1964 .

[22]  Lars Grasedyck,et al.  Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation , 2004, Numer. Linear Algebra Appl..

[23]  Anna Persson,et al.  Multiscale Differential Riccati Equations for Linear Quadratic Regulator Problems , 2017, SIAM J. Sci. Comput..

[24]  N. Higham.  COMPUTING A NEAREST SYMMETRIC POSITIVE SEMIDEFINITE MATRIX , 1988 .

[25]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[26]  Hermann Mena,et al.  On the benefits of the LDLT factorization for large-scale differential matrix equation solvers , 2015 .

[27]  W. Hundsdorfer,et al.  Modified Douglas splitting methods for reaction–diffusion equations , 2015, 1512.01445.

[28]  Thilo Penzl.  Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case , 2000 .

[29]  Willem Hundsdorfer,et al.  Accuracy and stability of splitting with stabilizing corrections , 2002 .

[30]  Luca Dieci,et al.  Numerical integration of the differential Riccati equation and some related issues , 1992 .

[31]  O. Jacobs,et al.  Introduction to Control Theory , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[32]  Peter Benner,et al.  Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations , 2018, J. Num. Math..

[33]  Tony Stillfjord,et al.  Singular Value Decay of Operator-Valued Differential Lyapunov and Riccati Equations , 2018, SIAM J. Control. Optim..