Efficient Solutions of Generalized k-Way Lyapunov Equations and their Application to Control of Nonlinear Systems
Year: 2026
Type: dissertation
Abstract: Nonlinear systems of differential equations often arise from the discretization of PDEs or modeling complex processes. Model reduction or optimal control of these systems are essential for numerous applications that require simulation and/or stabilization of these systems. Energy functions play a central role when performing model reduction of nonlinear systems using balanced truncation. On the other hand, the optimal control of problems with nonlinear constraints can be expressed through a so-called value function. The value function and energy functions can each be characterized as the solution of the Hamilton-Jacobi-Bellman (HJB) equations. We propose a polynomial approximation framework for HJB equations where the nonlinear dynamics are described by Stokes-type differential algebraic equations (DAEs). The polynomial coefficients require solutions to either k-way Lyapunov equations or generalized k-way Lyapunov equations. For the case where the systems are described by sparse large matrices, we propose preconditioning techniques that enable the application of standard iterative solvers for two classes of problems. The first involves matrices that can be expressed using Kronecker products of symmetric tri-diagonal systems. The second is more generally applicable, but requires the linear term in the dynamics to be stable. For these systems, we developed a recursive ADI method for k-way Lyapunov equations that is more attractive than existing algorithms when some specific original system matrices or the cost matrix are low rank. The proposed method significantly speeds up the simulations and provides accurate low-rank approximations of the solution of k-way Lyapunov equations. The effectiveness of the derived algorithms are demonstrated on benchmark systems.
Source: VTechWorks (Virginia Tech)
Author: Adjerid, Hamza
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203852913