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

TL;DR

This paper establishes new sufficient conditions ensuring bounded reachability sets in time-delay systems with mixed discrete and distributed delays, extending known results from finite-dimensional to more complex infinite-dimensional systems.

Contribution

It introduces novel criteria for boundedness of reachability sets in systems with mixed delays, applicable to a broad class of infinite-dimensional time-delay systems.

Findings

Derived sufficient conditions for bounded reachability sets

Identified broad classes of systems satisfying these conditions

Extended finite-dimensional results to systems with mixed delays

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.