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Eigensets of switching dynamical systems

Vladimir Protasov

Abstract

Reachability sets of linear switching dynamical systems (systems of ODE with time-dependent matrices that take values from a given compact set) are analysed. An eigenset is a non-trivial compact set M that possesses the following property: the closure of the set of points reachable by trajectories starting in M in time t is equal to exp(at)M. This concept introduced in a recent paper of E.Viscovini is an analogue of an eigenvector for compact sets of matrices. We prove the existence of eigensets, analyse their structure and properties, and find ``eigenvalues'' a for an arbitrary system. The question which compact sets, in particular, which convex sets and polyhedra, can be presented as eigensets of suitable systems, is studied.

Eigensets of switching dynamical systems

Abstract

Reachability sets of linear switching dynamical systems (systems of ODE with time-dependent matrices that take values from a given compact set) are analysed. An eigenset is a non-trivial compact set M that possesses the following property: the closure of the set of points reachable by trajectories starting in M in time t is equal to exp(at)M. This concept introduced in a recent paper of E.Viscovini is an analogue of an eigenvector for compact sets of matrices. We prove the existence of eigensets, analyse their structure and properties, and find ``eigenvalues'' a for an arbitrary system. The question which compact sets, in particular, which convex sets and polyhedra, can be presented as eigensets of suitable systems, is studied.
Paper Structure (15 theorems, 8 equations, 1 figure)

This paper contains 15 theorems, 8 equations, 1 figure.

Key Result

Theorem 1

For every eigenset of an irreducible system, we have ${\alpha = \sigma({\cal{A}})}$.

Figures (1)

  • Figure 1: Construction of eigensets of the system $\{A_1, A_2\}$

Theorems & Definitions (23)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Definition 3
  • Proposition 1
  • ...and 13 more