Yunfei Song

56495850000

Publications - 4

Invariance Preserving Discretization Methods of Dynamical Systems

Yunfei Song Tamás Terlaky Zoltán Horváth

Publication Name: Vietnam Journal of Mathematics

Publication Date: 2018-12-01

Volume: 46

Issue: 4

Page Range: 803-823

Description:

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 proven 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.

Open Access: Yes

DOI: 10.1007/s10013-018-0305-z

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

Yunfei Song Tamás Terlaky Zoltán Horváth

Publication Name: Applied Mathematics and Computation

Publication Date: 2017-04-01

Volume: 298

Issue: Unknown

Page Range: 351-367

Description:

In this paper, we propose a novel, simple, and unified approach to explore sufficient and necessary conditions, i.e., invariance conditions, under which four classic families of convex sets, namely, polyhedra, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for a linear discrete or continuous dynamical system. For discrete dynamical systems, we use the Theorems of Alternatives, i.e., Farkas lemma and S-lemma, to obtain simple and general proofs to derive invariance conditions. This novel method establishes a solid connection between optimization theory and dynamical system. Also, using the S-lemma allows us to extend invariance conditions to any set represented by a quadratic inequality. Such sets include nonconvex and unbounded sets. For continuous dynamical systems, we use the forward or backward Euler method to obtain the corresponding discrete dynamical systems while preserves invariance. This enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems. Finally, some numerical examples are presented to illustrate these invariance conditions.

Open Access: Yes

DOI: 10.1016/j.amc.2016.10.007

Invariance conditions for nonlinear dynamical systems

Yunfei Song Tamás Terlaky Zoltán Horváth

Publication Name: Springer Optimization and Its Applications

Publication Date: 2016-01-01

Volume: 115

Issue: Unknown

Page Range: 265-280

Description:

Recently,Horváth et al. (Appl Math Comput,submitted) proposed a novel unified approach to study,i.e.,invariance conditions,sufficient and necessary conditions,under which some convex sets are invariant sets for linear dynamical systems. In this paper,by utilizing analogous methodology,we generalize the results for nonlinear dynamical systems. First,the Theorems of Alternatives,i.e.,the nonlinear Farkas lemma and the S-lemma,together with Nagumo’s Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets,including polyhedral and ellipsoidal sets. Second,we establish an optimization framework to computationally verify the derived invariance conditions. Finally,we derive analogous invariance conditions without any conditions.

Open Access: Yes

DOI: 10.1007/978-3-319-42056-1_8

Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron

Yunfei Song Tamás Terlaky Zoltán Horváth

Publication Name: Discrete and Continuous Dynamical Systems Series A

Publication Date: 2015-07-01

Volume: 35

Issue: 7

Page Range: 2997-3013

Description:

Steplength thresholds for invariance preserving of three types of discretization methods on a polyhedron are considered. For Taylor approximation type discretization methods we prove that a valid steplength threshold can be obtained by finding the first positive zeros of a finite number of polynomial functions. Further, a simple and efficient algorithm is proposed to numerically compute the steplength threshold. For rational function type discretization methods we derive a valid steplength threshold for invariance preserving, which can be computed by using an analogous algorithm as in the first case. The relationship between the previous two types of discretization methods and the forward Euler method is studied. Finally, we show that, for the forward Euler method, the largest steplength threshold for invariance preserving can be computed by solving a finite number of linear optimization problems.

Open Access: Yes

DOI: 10.3934/dcds.2015.35.2997