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 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.
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Notes
In particular, this condition requires that if xk is in the set, then xk+ 1 is in the interior of the set.
A Lorenz cone can also be referred to as an ice cream cone, or a second-order cone.
Inertia{Q} = {a, b, c} means that matrix Q has a positive eigenvalues, b zero eigenvalues, and c negative eigenvalues.
A point \(x\in \mathcal {S}\) is called a relative interior point of \(\mathcal {S}\) if x is an interior point of \(\mathcal {S}\) relative to \(\text {aff}(\mathcal {S})\), where \(\text {aff}(\mathcal {S})\) is the smallest affine subspace containing \(\mathcal {S}\). Then, \(\text {ri}(\mathcal {S})\) is defined as the set of all the relative interior points of \(\mathcal {S}\), and \(\text {rb}(\mathcal {S})\) is defined as \(\text {cl}(\mathcal {S})\backslash \text {ri}(\mathcal {S})\) (see [23, p. 44]).
In practice, a possible way to obtain a base can be chosen as follows: we first take a hyperplane through the origin that intersects \(\mathcal {C}\) only by the origin. Then, shift the hyperplane to x∗, where x∗ is an interior point of \(\mathcal {C}\). The intersection of the shifted hyperplane and \(\mathcal {C}\) is a base of \(\mathcal {C}\). The base of \(\mathcal {C}\) is a compact set.
The dual of cone \(\mathcal {C}\) is defined as \(\mathcal {C}^{\ast }=\{y|y^{T}x\geq 0\text { for all } x\in \mathcal {C}\}\).
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Acknowledgements
The authors are grateful to the anonymous referees for carefully evaluating our manuscript, and for providing constructive suggestions which notably improved this paper.
Funding
This research is supported by a start-up grant of Lehigh University and by TAMOP-4.2.2.A-11/1KONV-2012-0012: Basic research for the development of hybrid and electric vehicles. The TAMOP Project is supported by the European Union and co-financed by the European Regional Development Fund.
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Dedicated to Professor Hans Georg Bock on the occasion of his 70th birthday.
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Horváth, Z., Song, Y. & Terlaky, T. Invariance Preserving Discretization Methods of Dynamical Systems. Vietnam J. Math. 46, 803–823 (2018). https://doi.org/10.1007/s10013-018-0305-z
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DOI: https://doi.org/10.1007/s10013-018-0305-z