[Paper Review] Hybrid systems with memory: Existence and well-posedness of generalized solutions

Jun Liu, Andrew R. Teel|arXiv (Cornell University)|Jul 20, 2015

Stability and Controllability of Differential Equations3 citations

TL;DR

This paper establishes a theoretical framework for hybrid systems with memory using generalized solutions on hybrid time domains, employing graphical convergence topology to ensure existence and well-posedness. The key contribution is proving that pre-asymptotic stability in such systems is robust under perturbations, enabling a unified analysis of discrete and continuous-time systems with delays.

ABSTRACT

Hybrid systems with memory refer to dynamical systems exhibiting both hybrid and delay phenomena. While systems of this type are frequently encountered in many physical and engineering systems, particularly in control applications, various issues centered around the robustness of hybrid delay systems have not been adequately dealt with. In this paper, we establish some basic results on a framework that allows to study hybrid systems with memory through generalized concepts of solutions. In particular, we develop the basic existence of generalized solutions using regularity conditions on the hybrid data, which are formulated in a phase space of hybrid trajectories equipped with the graphical convergence topology. In contrast with the uniform convergence topology that has been often used, adopting the graphical convergence topology allows us to establish well-posedness of hybrid systems with memory. We then show that, as a consequence of well-posedness, pre-asymptotic stability of well-posed hybrid systems with memory is robust.

Motivation & Objective

• To address the lack of robust stability analysis tools for hybrid systems with memory, particularly due to discontinuities from jumps and delays.
• To extend generalized solution concepts—previously used for hybrid systems without memory—to systems with functional delays.
• To develop a well-posedness theory that supports robust stability analysis under perturbations of hybrid data.
• To unify the analysis of discrete-time and continuous-time systems with delays through a common theoretical framework.
• To provide foundational results for robust stability theory in hybrid systems with memory, supporting future developments in control applications.

Proposed method

• Formulates hybrid systems with memory using generalized solutions defined on hybrid time domains parameterized by real time and jump count.
• Employs the graphical convergence topology on a phase space of hybrid trajectories, replacing the standard uniform convergence topology to better handle solution discontinuities.
• Establishes existence of generalized solutions via regularity conditions on hybrid data, including flow and jump maps defined on the space of memory arcs.
• Introduces perturbations of hybrid data and defines well-posedness in terms of solution existence, uniqueness, and continuous dependence on initial conditions.
• Uses tools from functional differential inclusions and set-valued analysis to prove well-posedness under mild regularity assumptions.
• Applies the well-posedness result to prove robustness of pre-asymptotic stability, showing that stability is preserved under small perturbations of the system data.

Experimental results

Research questions

• RQ1 How can generalized solutions be defined and proven to exist for hybrid systems with memory, given the challenges posed by delays and jumps?
• RQ2 What topological framework enables robust analysis of hybrid systems with memory, particularly when standard uniform convergence fails?
• RQ3 Under what conditions is the solution set of a hybrid system with memory well-posed, ensuring existence, uniqueness, and continuous dependence?
• RQ4 Is pre-asymptotic stability in hybrid systems with memory robust to perturbations of the system data?
• RQ5 Can the proposed framework unify the analysis of discrete-time and continuous-time systems with delays in a single theoretical setting?

Key findings

• Generalized solutions for hybrid systems with memory exist under mild regularity conditions on the hybrid data, including flow and jump sets defined on the space of memory arcs.
• Well-posedness of hybrid systems with memory is established using the graphical convergence topology, which better captures structural properties of solutions than uniform convergence.
• Pre-asymptotic stability for well-posed hybrid systems with memory is robust under perturbations of the system data, ensuring stability is preserved under small disturbances.
• The robustness of pre-asymptotic stability is formally proven via a contradiction argument using graphical convergence of solution sequences and the limiting solution's stability properties.
• The framework enables the extension of key stability tools—such as converse Lyapunov theorems and invariance principles—to hybrid systems with memory.
• The results support the development of a comprehensive robust stability theory for hybrid systems with delays, unifying discrete and continuous dynamics with memory effects.