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Time-Delay Systems with Discrete and Distributed delays: Discontinuous Initial Conditions and Reachability Sets

Hernan Haimovich, Jose L. Mancilla-Aguilar

Abstract

Time-invariant finite-dimensional systems, under reasonable continuity assumptions, exhibit the property that if solutions exist for all future times, the set of vectors reachable from a bounded set of initial conditions over bounded time intervals is also bounded. This property can be summarized as follows: forward completeness implies bounded reachability sets. By contrast, this property does not necessarily hold for infinite-dimensional systems in general, and time-delay systems in particular. Sufficient conditions for this property to hold that can be directly tested on the function defining the system dynamics are only known in the case of systems with pointwise (or discrete) delays. This paper develops novel sufficient conditions for the boundedness of the reachability sets of time-delay systems involving mixed pointwise and distributed delays. Broad classes of systems satisfying these conditions are identified.

Time-Delay Systems with Discrete and Distributed delays: Discontinuous Initial Conditions and Reachability Sets

Abstract

Time-invariant finite-dimensional systems, under reasonable continuity assumptions, exhibit the property that if solutions exist for all future times, the set of vectors reachable from a bounded set of initial conditions over bounded time intervals is also bounded. This property can be summarized as follows: forward completeness implies bounded reachability sets. By contrast, this property does not necessarily hold for infinite-dimensional systems in general, and time-delay systems in particular. Sufficient conditions for this property to hold that can be directly tested on the function defining the system dynamics are only known in the case of systems with pointwise (or discrete) delays. This paper develops novel sufficient conditions for the boundedness of the reachability sets of time-delay systems involving mixed pointwise and distributed delays. Broad classes of systems satisfying these conditions are identified.
Paper Structure (17 sections, 14 theorems, 95 equations)

This paper contains 17 sections, 14 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Time-delay Systems with Mixed Delays
  3. Notation
  4. System, Existence and Uniqueness
  5. Time-delay Systems
  6. Existence and Uniqueness of Solutions
  7. Combining Discrete and Distributed Delays
  8. Equivalent Initial Conditions
  9. Forward Completeness and Bounded Reachability Sets
  10. Weak-* Topology
  11. Uniform Convergence of Solutions
  12. Bounded Reachability Sets
  13. Discussion
  14. Conclusions
  15. Some Technical Proofs
  16. ...and 2 more sections

Key Result

Proposition 2.3

Let $\mathbf{f}$ satisfy Assumption as:Lip. Then, for each $t_0\ge 0$ and $\phi\in \mathcal{L}^\infty$ with $\|\phi\|_{\infty} < \infty$, there is a unique maximally defined solution $x : [t_0,T_x) \to {\mathbb R}^n$ of (eq:td-gen). Moreover, if $\sup_{t_0 \le t < T_x} |x(t)| < \infty$, then $T_x =

Theorems & Definitions (33)

  • Remark 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • Remark 2.10
  • ...and 23 more