⚡ electrical engineering

Asymptotic Stability and Decay Rates of Homogeneous Positive Systems With Bounded and Unbounded Delays

This paper establishes necessary and sufficient conditions for the delay-independent asymptotic stability of continuous- and discrete-time homogeneous positive systems with time-varying (including unbounded) delays, while also deriving explicit expressions for their decay rates.

Original authors: Hamid Reza Feyzmahdavian, Themistoklis Charalambous, Mikael Johansson

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 3.0

Original authors: Hamid Reza Feyzmahdavian, Themistoklis Charalambous, Mikael Johansson

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a bustling city where information flows between different neighborhoods. In this city, the "state" of the system is like the population or traffic in each neighborhood. Usually, in real-world systems like power grids, biological populations, or economic markets, these quantities can't be negative—you can't have -5 people or -10 dollars. These are called Positive Systems.

Now, imagine that sending a message from one neighborhood to another takes time. Sometimes the mail is fast, sometimes slow, and sometimes the delay changes unpredictably. This is a Time-Delay System.

For a long time, scientists knew that if the delays were fixed (like a constant 5-minute wait), these positive systems were surprisingly robust. If the city was stable without delays, it stayed stable even with fixed delays. But what if the delays were changing, or even growing infinitely long? And what if the rules governing the city weren't simple straight lines (linear) but complex curves (nonlinear)?

This paper, by Feyzmahdavian, Charalambous, and Johansson, tackles exactly these messy, real-world scenarios. Here is the breakdown of their findings using simple analogies:

1. The "Insensitivity" Superpower

The authors study a specific type of city where the rules are Cooperative (neighborhoods help each other grow) and Homogeneous (the rules scale up or down proportionally, like a recipe that works whether you feed 10 people or 1,000).

They discovered a "superpower" of these systems: Delay Independence.

The Analogy: Imagine a group of friends trying to coordinate a meeting. If the group is naturally cooperative and follows a specific scaling rule, it doesn't matter if the text messages take 1 second or 1 hour to arrive, or if the wait time fluctuates wildly.
The Finding: If the system is stable when everyone talks instantly (no delays), it will remain stable even if the delays are changing, unpredictable, or even growing infinitely long (as long as the old information eventually gets "purged" from the system). You don't need to know exactly how long the delay is to know the system won't crash.

2. The "Speed Limit" of Recovery

While the system won't crash (it's stable), the paper asks: How fast does it recover from a shock?

The Analogy: If a storm hits the city, how quickly does the traffic return to normal?
The Finding: The speed of recovery does depend on the delays.
- Bounded Delays (The Short Wait): If the delay is capped at a maximum time (e.g., "no message takes longer than 10 minutes"), the system recovers quickly.
  - If the rules are simple (degree 0), it recovers exponentially fast (like a ball bouncing back up quickly).
  - If the rules are complex (degree > 0), it recovers at a polynomial rate (slower, but still steady).
- Unbounded Delays (The Growing Wait): If the delay can grow forever (e.g., "the wait time gets longer every day"), the recovery slows down significantly.
  - The paper provides a mathematical "speed limit" showing that as the delays grow faster, the system's recovery rate slows down, eventually becoming a slow, power-law decay rather than a fast exponential one.

3. Continuous vs. Discrete Time

The paper covers two types of time:

Continuous Time: Like water flowing in a river (e.g., power control in a wireless network). The authors show that for these systems, the "insensitivity" holds true for any degree of complexity.
Discrete Time: Like a digital clock ticking second by second (e.g., a computer updating a database).
- For simple rules (degree 0), the same "insensitivity" applies.
- For complex rules (degree > 0), the system is only locally stable.
- The Analogy: Think of a tightrope walker. If the rules are simple, they can walk the rope even if the wind (delay) is crazy. If the rules are complex, they can only stay balanced if they start very close to the center; if they start too far out, the crazy wind might knock them off.

4. The "Recipe" for Stability

How do you check if your system is safe? The authors provide a simple test (a "recipe"):

Find a specific set of positive numbers (a vector v).
Plug these numbers into the system's equations.
If the result is negative (meaning the system naturally wants to shrink back to zero), then you are safe.
The Magic: You don't need to know the delay. You just check the system without delays. If it passes the test, it passes for any delay that satisfies the basic condition of eventually letting old information fade away.

Summary

This paper proves that for a wide class of positive systems (where quantities stay non-negative), stability is robust against time delays. Whether the delays are constant, changing, or growing infinitely, the system won't blow up if it's stable without delays. However, the speed at which it settles down depends heavily on how fast those delays are growing. The authors provide the exact formulas to calculate this speed, helping engineers and scientists predict how fast their systems will recover in the real world.

Technical Summary: Asymptotic Stability and Decay Rates of Homogeneous Positive Systems With Bounded and Unbounded Delays

Problem Statement

The paper addresses the stability analysis of nonlinear positive systems in the presence of time delays. While significant literature exists on the stability of positive linear systems with delays, the theory for nonlinear positive systems is less developed. Existing results for nonlinear systems often rely on restrictive assumptions, such as constant delays or specific degrees of homogeneity (e.g., degree zero). Furthermore, many studies assume delays are bounded, whereas real-world applications (such as wireless power control) may involve time-varying delays that are unbounded, provided they grow slower than time itself.

The core problem is to establish necessary and sufficient conditions for the delay-independent global asymptotic stability of a broad class of nonlinear positive systems (specifically, those with cooperative and homogeneous vector fields in continuous time, and non-decreasing and homogeneous vector fields in discrete time). Additionally, the paper seeks to quantify the decay rates of these systems under various classes of time-varying delays, including both bounded and unbounded cases.

Methodology

The authors employ a proof technique distinct from the widely used Lyapunov-Krasovskii functional method (common for constant delays) and the monotonicity-based approaches used in prior work on constant delays. Instead, the methodology relies on constructing a specific Lyapunov function and analyzing the system's trajectory through nested sets.

Key Methodological Elements:

System Class: The study focuses on systems where the vector fields are:
- Continuous-time: Cooperative (Jacobian is Metzler) and homogeneous of arbitrary degree $p$ .
- Discrete-time: Non-decreasing and homogeneous of degree zero (for global stability) or degree $p > 0$ (for local stability).
Delay Assumptions: The paper considers time-varying delays $\tau(t)$ (continuous) or $d(k)$ (discrete) satisfying a minimal growth condition: $\lim_{t \to \infty} (t - \tau(t)) = +\infty$ . This allows for both bounded delays and unbounded delays that do not "outrun" time.
Lyapunov Construction: A weighted $l_\infty$ -norm based Lyapunov function is defined:
$V(x) = \max_{1 \le i \le n} \left( \frac{x_i}{v_i} \right)^{\frac{r_{\max}}{r_i}}$
where $v > 0$ is a specific vector and $r$ defines the dilation map.
Nested Set Argument: The proof demonstrates that if a stability condition is met, the system state enters a sequence of nested sets $S(m)$ defined by $V(x) \le \gamma^m \|\phi\|$ (where $0 \le \gamma < 1$ ) and remains there. This establishes that the state converges to the origin regardless of the delay magnitude or variation, provided the delay satisfies the growth assumption.
Decay Rate Analysis: To quantify convergence, the authors utilize the concept of $\mu$ -stability. They derive conditions under which the solution satisfies $\|x(t)\| = O(\mu^{-1}(t))$ , where $\mu(t)$ is a non-decreasing function characterizing the decay rate (e.g., exponential or polynomial).

Key Contributions and Results

1. Delay-Independent Stability Conditions

The paper establishes that for the specified class of systems, global asymptotic stability is independent of the magnitude and variation of time delays, provided the delays satisfy the growth condition (Assumption 1 for continuous-time, Assumption 4 for discrete-time).

Continuous-Time (Theorem 1): A homogeneous cooperative system is globally asymptotically stable for all admissible delays if and only if there exists a vector $v > 0$ such that:
$f(v) + g(v) < 0$
This condition is identical to the stability condition for the corresponding delay-free system.
Discrete-Time (Theorem 3): For homogeneous non-decreasing systems of degree zero, global asymptotic stability for all admissible delays holds if and only if there exists $v > 0$ such that:
$f(v) + g(v) < v$
Local Stability for Higher Degrees: For discrete-time systems with degree $p > 0$ , the same condition ensures local asymptotic stability for initial conditions within a specific bound.

2. Quantification of Decay Rates

The paper provides explicit bounds on decay rates for different delay scenarios:

Bounded Delays (Corollary 1):
- If the degree $p=0$ , the system is globally exponentially stable. The decay rate $\eta$ is determined by a transcendental equation involving the delay bound $\tau_{\max}$ .
- If the degree $p>0$ , the system exhibits power-rate stability (polynomial decay), with the rate depending on $\tau_{\max}$ .
Unbounded Delays (Corollary 2 & 5):
- For delays satisfying $\sup_{t>T} \frac{\tau(t)}{t} = \alpha < 1$ , the system remains power-rate stable.
- The decay rate is shown to be of the order $O(t^{-\xi})$ (for $p=0$ ) or $O(t^{-\frac{r_{\max}}{p}\beta})$ (for $p>0$ ).
- Crucially, the paper quantifies how the parameter $\alpha$ (the asymptotic growth rate of the delay) impacts stability: as $\alpha \to 1$ , the guaranteed decay rate deteriorates (approaches zero), indicating slower convergence.

3. Extension to Linear Systems

The results generalize existing findings for positive linear systems.

For continuous-time linear systems ( $\dot{x} = Ax + Bx(t-\tau(t))$ ), stability is guaranteed if $A+B$ is Hurwitz (all eigenvalues have negative real parts).
For discrete-time linear systems ($x(k+1) = Ax(k) + Bx(k-d(k))$), stability is guaranteed if the spectral radius of $A+B$ is less than 1.
Unlike previous works that only established stability, this paper provides explicit decay rate bounds for these linear systems under unbounded delays.

Significance and Claims

The paper claims to provide a comprehensive framework for analyzing the stability of a significant class of nonlinear positive systems under minimal delay restrictions.

Generality: The results extend beyond constant delays to time-varying and unbounded delays, a scenario where previous monotonicity-based proofs failed.
Minimal Restrictions: The proof technique imposes minimal restrictions on the delays, requiring only that they do not grow faster than time itself.
Quantitative Insight: A major contribution is the derivation of explicit decay rate bounds. The paper demonstrates that while stability is delay-independent, the speed of convergence is not; it is directly influenced by the growth rate of the delays.
Unification: The work unifies the treatment of continuous and discrete-time systems and bridges the gap between linear and nonlinear positive systems regarding delay insensitivity.

The authors conclude that their findings offer a robust theoretical basis for the analysis and control of positive systems in distributed environments where information exchange delays are inevitable and potentially unbounded.