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Solution Concepts and Existence Results for Hybrid Systems with Continuous-time Inputs

W. P. M. H. Heemels, R. Postoyan, P. Bernard, K. J. A. Scheres, R. G. Sanfelice

TL;DR

This work addresses the existence and forward completeness of solutions for hybrid dynamical inclusions with exogenous continuous-time inputs, introducing e-solution and ae-solution concepts under a fixed input signal. It develops trajectory-dependent viability conditions VC-e and VC-ae and then translates these into trajectory-independent tangent-cone conditions, covering absolutely continuous, continuous, and measurable inputs. The results include a novel treatment of input-measurable cases and a rigorous handling of càdlàg inputs, with propositions guaranteeing the existence of nontrivial solutions and characterizing maximal-solution behavior. The findings advance well-posedness analysis for hybrid control, estimation, and disturbance rejection in systems where external signals influence flow and jump dynamics, and connect to viability theory for non-autonomous inclusions.

Abstract

In many scenarios, it is natural to model a plant's dynamical behavior using a hybrid dynamical system influenced by exogenous continuous-time inputs. While solution concepts and analytical tools for existence and completeness are well established for autonomous hybrid systems, corresponding results for hybrid dynamical systems involving continuous-time inputs are generally lacking. This work aims to address this gap. We first formalize notions of a solution for such systems. We then provide conditions that guarantee the existence and forward completeness of solutions. Moreover, we leverage results and ideas from viability theory to present more explicit conditions in terms of various tangent cone formulations. Variants are provided that depend on the regularity of the exogenous input signals.

Solution Concepts and Existence Results for Hybrid Systems with Continuous-time Inputs

TL;DR

This work addresses the existence and forward completeness of solutions for hybrid dynamical inclusions with exogenous continuous-time inputs, introducing e-solution and ae-solution concepts under a fixed input signal. It develops trajectory-dependent viability conditions VC-e and VC-ae and then translates these into trajectory-independent tangent-cone conditions, covering absolutely continuous, continuous, and measurable inputs. The results include a novel treatment of input-measurable cases and a rigorous handling of càdlàg inputs, with propositions guaranteeing the existence of nontrivial solutions and characterizing maximal-solution behavior. The findings advance well-posedness analysis for hybrid control, estimation, and disturbance rejection in systems where external signals influence flow and jump dynamics, and connect to viability theory for non-autonomous inclusions.

Abstract

In many scenarios, it is natural to model a plant's dynamical behavior using a hybrid dynamical system influenced by exogenous continuous-time inputs. While solution concepts and analytical tools for existence and completeness are well established for autonomous hybrid systems, corresponding results for hybrid dynamical systems involving continuous-time inputs are generally lacking. This work aims to address this gap. We first formalize notions of a solution for such systems. We then provide conditions that guarantee the existence and forward completeness of solutions. Moreover, we leverage results and ideas from viability theory to present more explicit conditions in terms of various tangent cone formulations. Variants are provided that depend on the regularity of the exogenous input signals.
Paper Structure (17 sections, 9 theorems, 19 equations)

This paper contains 17 sections, 9 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Solution concepts
  3. Class of systems and preliminaries
  4. Definitions
  5. Existence and completeness of solutions
  6. Existence of non-trivial solutions
  7. Properties of maximal solutions
  8. Examples
  9. Restricting the inputs to càdlàg signals
  10. Trajectory-independent viability conditions
  11. Absolutely continuous and continuous inputs
  12. Absolutely continuous inputs
  13. Continuous inputs
  14. Measurable inputs
  15. Measurable inputs not affecting $C$
  16. ...and 2 more sections

Key Result

Proposition 3.1

Consider the hybrid system $\mathcal{H}=(C,F,D,G,W)$. (i) [one initial state, one input] There exists a nontrivial e-solution $\phi$ to $\mathcal{H}$ with input $w\in{\mathcal{L}}_{W}$ and $\phi(0,0)=\xi\in\mathbb{R}^{n_x}$ if and only if $(\xi,w(0))\in D$ or VC-e($\xi,w$) holds. (ii) [multiple in

Theorems & Definitions (15)

  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Example 3.1
  • Example 3.2
  • Proposition 4.1
  • Example 4.1
  • Remark 4.1
  • Proposition 5.1
  • Remark 5.1
  • ...and 5 more