Construction of an iterative method for solving a class of complex symmetric generalized Lyapunov matrix equation and application to Helmholtz equation

Shirilord, Akbar; Dehghan, Mehdi

doi:10.22060/ajmc.2025.24112.1366

Construction of an iterative method for solving a class of complex symmetric generalized Lyapunov matrix equation and application to Helmholtz equation

Articles in Press

Document Type : Original Article

Authors

Akbar Shirilord

Mehdi Dehghan

Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran

10.22060/ajmc.2025.24112.1366

Abstract

The Lyapunov matrix equations occur in many branches of control theory, such as stability analysis and optimal control. In this work, we introduce a novel iterative approach to address the generalized Lyapunov matrix equation within the framework of complex matrices. The procedure involves solving two conventional Lyapunov equations with real-valued coefficient matrices at each iteration. The scheme incorporates two positive parameters, for which we establish sufficient conditions to guarantee the convergence of the method under certain assumptions. Then we solve the Lyapunov equation arising from applying a finite difference procedure to the Helmholtz equation by the proposed method.

Keywords

Iterative schemes

Generalized Lyapunov equation

Control theory and systems control framework

Helmholtz equation

Convergence condition

Subjects