Application of the implicit Euler method for the discretization of some classes of nonlinear systems

A. Yu. Aleksandrov

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 19 (2023) no. 3, pp. 304-319

Résumé

The problem of stability preservation under discretization of some classes of nonlinear differential equations systems is studied. Persidskii systems, Lurie systems of indirect control, and systems whose right-hand sides have a canonical structure are considered. It is assumed that the zero solutions of these systems are globally asymptotically stable. Conditions are determined that guarantee the asymptotic stability of the zero solutions for the corresponding difference systems. Previously, such conditions were established for the case where discretization was carried out using the explicit Euler method. In this paper, difference schemes are constructed on the basis of the implicit Euler method. For the obtained discrete systems, theorems on local and global asymptotic stability are proved, estimates of the time of transient processes are derived. For systems with a canonical structure of right-hand sides, based on the approach of V. I. Zubov, a modified implicit computational scheme is proposed that ensures the matching of the convergence rate of solutions to the origin for the differential and corresponding difference systems. It is shown that implicit computational schemes can guarantee the preservation of asymptotic stability under less stringent constraints on the discretization step and right-hand sides of the systems under consideration compared to the constraints obtained using the explicit method. An example is presented illustrating the obtained theoretical conclusions.

Détail

Keywords: difference systems, discretization, asymptotic stability, Lyapunov functions, conservative numerical methods.
Mots-clés : implicit Euler method