Invariance Preserving Discretization Methods of Dynamical Systems

Abstract

In this paper, we consider local and uniform invariance preserving steplength thresholds on a set when a discretization method is applied to a linear or nonlinear dynamical system. For the forward or backward Euler method, the existence of local and uniform invariance preserving steplength thresholds is proved when the invariant sets are polyhedra, ellipsoids, or Lorenz cones. Further, we also quantify the steplength thresholds of the backward Euler methods on these sets for linear dynamical systems. Finally, we present our main results on the existence of uniform invariance preserving steplength threshold of general discretization methods on general convex sets, compact sets, and proper cones both for linear and nonlinear dynamical systems. ZOLTÁN HORVÁTH, YUNFEI SONG AND TAMÁS TERLAKY sponding discrete system which is obtained by using the discretization method. Such a discretization method is called invariance preserving for the dynamical system on the positively invariant set.In this paper, our focus is to find conditions, in particular steplength thresholds for the discretization methods, such that the considered discretization method is invariance preserving for the given linear or nonlinear dynamical system. This topic is of great interest in the fields of dynamical systems, partial differential equations, and control theory. A basic result is presented in Bolley & Crouzeix (1978), which considers linear problems and invariance preserving on the positive orthant from a perspective of numerical methods. For invariance preserving on the positive orthant or polyhedron for Runge-Kutta methods, the reader is refereed to Horváth (2004); Horváth (2005, 2006). A similar concept named strong stability preserving (SSP) used in numerical methods is studied in Gottlieb et al. (2011Gottlieb et al. ( , 2001; Shu & Osher (1988). These papers deal with invariance preserving of general sets and they usually use the assumption that the Euler methods are invariance preserving with a steplength threshold τ 0 . Then the uniform invariance preserving steplength threshold for other advanced numerical methods, e.g., Runge-Kutta methods, is derived in terms of τ 0 . Therefore, to make the results applicable to solve real world problems, this approach requires to check whether such a positive τ 0 exists for Euler methods. We note that quantifications of steplength thresholds for some classes discretization methods on a polyhedron are studied in Horváth et al. (2015).In this paper, basic concepts and theorems are introduced in Section 2. In Section 3 first we prove that for the forward Euler method, a local invariance preserving steplength threshold exists for a given polyhedron when a linear dynamical system is considered. For the backward Euler method we prove that a local steplength threshold exists for polyhedron, ellipsoid, and Lorenz cone. These proofs are using elementary concepts. We also quantify a valid local steplength threshold for the backward Euler method. Second, we prove that a uniform invariance preserving steplengt...

Invariance Conditions for Nonlinear Dynamical Systems

“…Theorem 5.5. Let the convex set S be given as in (10) and let the discrete system be given as in (1). Then S is an invariant set for the discrete system if and only if there exists an α ≥ 0, such that αg(x) − g(f d (x)) ≥ 0, for all x ∈ S.…”

“…Theorem 4.5. Let the convex set S be given as in (10) and let function g(x) be continuously differentiable. Then S is an invariant set for the continuous system (2)…”

“…Invariant set for discrete system is studied in [14], and an application to model predictive control is provided. The steplength threshold for preserving invariance of a set when applying a discretization method to continuous systems is studied in [10,12].…”

A novel unified approach to invariance conditions for a linear dynamical system

“…Here we formally present these results as follows. The first statement can be found in [8,10,11,34], and the second statement can be found in [34].…”

“…According to [34], we have that both the forward and backward Euler methods are invariance preserving for a polyhedral set. Blanchini [8,10] presents the connection between invariant sets for continuous and discrete systems by using the forward Euler method.…”