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Concentration of Stochastic System Trajectories with Time-varying Contraction Conditions

This paper establishes two tight concentration inequalities for nonlinear stochastic systems under time-varying contraction conditions by introducing an Averaged Moment Generating Function (AMGF) combined with incremental stability and martingale methods to bound both single-time deviations and entire trajectory fluctuations with an $\mathcal{O}(\sqrt{\log(1/\delta)})$ guarantee.

Original authors: Zishun Liu, Liqian Ma, Hongzhe Yu, Yongxin Chen

Published 2026-04-03

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CC BY 4.0

Original authors: Zishun Liu, Liqian Ma, Hongzhe Yu, Yongxin Chen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 you are trying to guide a very energetic, slightly drunk friend (let's call him Stochastic Sam) through a crowded, obstacle-filled city to a specific destination.

The Plan: You have a perfect, ideal route drawn on a map. This is the Deterministic Trajectory (the path Sam would take if he were sober and perfectly focused).
The Reality: Sam is stumbling, getting bumped by people, and reacting to random gusts of wind. His actual path is the Stochastic Trajectory.
The Goal: You need to guarantee that Sam stays within a safe "bubble" around your ideal route so he doesn't crash into buildings or fall off cliffs. You want to say, "I am 99.99% sure Sam will stay inside this bubble."

This paper is about figuring out exactly how big that safety bubble needs to be, especially when the city layout (the rules of the road) is constantly changing.

The Old Way vs. The New Way

The Old Way (Incremental Stability Analysis):
Previously, mathematicians tried to predict Sam's path by looking at his average behavior. It was like saying, "On average, Sam stumbles 1 meter." To be safe, they would draw a massive bubble around the path—maybe 10 meters wide.

The Problem: This bubble was way too big. It was so conservative that it made it impossible to navigate tight spaces. If you needed to be 99.99% sure, the old math would demand a bubble so huge it covered the whole city block. It was safe, but useless for precision.

The New Way (This Paper):
The authors, Zishun Liu and his team, developed a smarter way to measure Sam's wobble. They introduced a new tool called the AMGF (Averaged Moment Generating Function).

Think of the AMGF as a "Super-Sensitive Energy Meter."
Instead of just measuring the average stumble, this meter measures the potential energy of Sam's wobble in every possible direction at once. It's like having a 360-degree sensor that knows exactly how much Sam might lean left, right, forward, or backward, and how that energy grows or shrinks over time.

The Two Main Tricks

The paper solves the problem in two steps, like building a safety net:

1. The "Snapshot" Safety Net (Single Time)
First, they figured out how to draw a tight bubble around Sam at any single moment in time.

The Magic: Because they used the AMGF, they found that the size of the bubble doesn't need to grow wildly just because you want higher confidence.
The Analogy: If the old method said, "To be 99.99% sure, you need a 10-meter bubble," the new method says, "To be 99.99% sure, you only need a 0.5-meter bubble." That's a huge difference! It's the difference between needing a giant warehouse to store a bicycle versus a simple bike rack.

2. The "Whole Journey" Safety Net (The Entire Trajectory)
The hard part is guaranteeing Sam stays safe for the entire trip, not just at one stop. If you just check him every second, you might miss a moment where he stumbles between checks.

The Martingale Method: The authors used a mathematical concept called a "Martingale" (think of it as a fair game or a balanced scale). They treated Sam's deviation from the path as a game where the odds are always fair. By using this, they could draw a continuous "tube" around the entire path that Sam is guaranteed to stay inside.
The "Strong Contraction" Upgrade: If the city is designed such that Sam is naturally pulled back toward the center (like a magnet), the safety tube can get even tighter. The authors combined their "Snapshot" and "Journey" methods to create a super-tight tube for these specific scenarios. They essentially chopped the journey into tiny slices, checked the safety at the start and end of each slice, and stitched them together. This prevents the safety bubble from ballooning out of control over long trips.

The Real-World Test: The Flying Robot

To prove this works, they tested it on a PVTOL (a flying robot that takes off and lands vertically, like a drone).

The Scenario: The drone had to fly through a maze of obstacles to reach a goal.
The Wind: Random wind gusts (noise) were blowing the drone off course.
The Result: Using their new math, they calculated a safety bubble that was small enough to fit through the maze.
- The old math would have said the bubble is so big (over 10 meters) that the drone would crash into the walls immediately.
- The new math calculated a bubble of about 0.54 meters.
The Outcome: They ran the simulation 10,000 times. In every single run, the drone stayed inside the small bubble and successfully reached the goal without hitting anything.

Why This Matters

In the real world, we are putting robots, self-driving cars, and power grids into chaotic environments. We can't just say "it's probably fine." We need mathematical guarantees.

This paper gives engineers a new, much sharper tool. It allows them to design systems that are:

Safer: They can prove with high confidence (99.99%) that the system won't fail.
More Efficient: Because the safety bubbles are smaller, robots can navigate tighter spaces and move faster without risking a crash.

In short: The authors invented a new "energy meter" that lets us draw much smaller, tighter safety nets around moving systems, ensuring they stay safe even when the world is chaotic and changing.

1. Problem Formulation

The paper addresses the safety verification of nonlinear stochastic systems subject to inherent random disturbances (modeled as Itô stochastic differential equations). The core problem is to quantify the deviation between a stochastic trajectory ( $X_t$ ) and its deterministic counterpart ( $x_t$ ) under time-varying contraction conditions.

The authors define two specific problems:

Problem 1 (Single-Time Concentration): Determine a tight probabilistic bound $r_{\delta,t}$ such that the deviation $\|X_t - x_t\|_{M_t}$ at a specific time $t$ is within the bound with probability $1-\delta$ .
Problem 2 (Trajectory-Level Concentration): Determine a time-varying tube $r_{\delta,t}$ that bounds the deviation for the entire trajectory over a time horizon $[0, T]$ with probability $1-\delta$ .

Key Challenge: Traditional methods based on Incremental Stability Analysis (ISA) often yield overly conservative bounds (scaling as $O(\sqrt{1/\delta})$ ) and struggle to characterize the entire trajectory without resorting to coarse time discretization. Furthermore, existing tight bounds were limited to time-invariant contraction conditions.

2. Methodology

The proposed approach integrates Contraction Theory, a novel energy function called the Averaged Moment Generating Function (AMGF), and Martingale-based methods.

A. The Averaged Moment Generating Function (AMGF)

Instead of bounding the quadratic expectation $E[\|X_t - x_t\|^2]$ , which leads to loose Markov inequalities, the authors utilize the AMGF, defined as:
$\Phi(X) = E_{\ell \in S^{n-1}} [e^{\lambda \langle \ell, X \rangle}]$
where $\ell$ is a random unit vector. The weighted version $\Phi_M(X)$ is used to measure deviation in the contraction metric $M_t$ .

Key Property: The AMGF allows the derivation of bounds that scale as $O(\sqrt{\log(1/\delta)})$ , which is significantly tighter than the $O(\sqrt{1/\delta})$ scaling of traditional methods, especially for high-confidence levels (small $\delta$ ).

B. Single-Time Bound (Theorem 1)

Using path-length integral techniques, the authors analyze the evolution of the AMGF along geodesics connecting the deterministic and stochastic trajectories.

They derive a differential inequality for the expected AMGF energy.
By solving this inequality, they establish a bound for the deviation at a single time instant $t$ .
Result: The bound scales with $\sqrt{\log(1/\delta)}$ , representing a fundamental improvement over ISA.

C. Trajectory-Level Bound (Theorem 2)

To extend the result to the entire trajectory $[0, T]$ , the authors employ Affine Martingales.

They prove that the AMGF of the deviation, when scaled by an exponential factor, acts as an affine martingale.
Using a lemma regarding the sublevel sets of affine martingales, they derive a tube that probabilistically envelopes the entire trajectory.
Limitation: For strongly contracting systems over long horizons, this bound can become conservative because the accumulated noise term grows exponentially with time.

D. Improved Bound for Strongly Contracting Systems (Theorem 3)

To address the conservativeness of Theorem 2 in strongly contracting systems ( $c_t < 0$ ), the authors propose a hybrid approach:

Time Discretization: Split the horizon $[0, T]$ into small intervals of length $\Delta t$ .
Hybrid Application: Apply Theorem 2 (trajectory bound) within each short interval and Theorem 1 (single-time bound) at the interval endpoints.
Union Bound: Combine the probabilities using a union bound.

Result: This method replaces the global exponential growth term with a local term dependent on $\Delta t$ , significantly sharpening the bound for large $T$ .

3. Key Contributions

Generalization to Time-Varying Conditions: The results generalize previous work (which was limited to time-invariant contraction) to systems with time-varying contraction metrics and rates.
Tightness of Bounds: The derived inequalities achieve an $O(\sqrt{\log(1/\delta)})$ dependence on the probability level $\delta$ . This is theoretically tight and significantly less conservative than the $O(\sqrt{1/\delta})$ dependence of standard ISA methods.
Trajectory-Level Analysis: The paper provides rigorous trajectory-level concentration inequalities without requiring coarse time discretization, ensuring safety over the continuous time domain.
Optimization for Strong Contraction: The synthesis of single-time and trajectory bounds (Theorem 3) offers a sharper solution for strongly contracting systems, overcoming the exponential blow-up of global bounds.

4. Results and Validation

Theoretical Validation: The paper proves that the derived bounds are the tightest obtainable under the given assumptions.
Case Study (PVTOL System):
- Scenario: A Planar Vertical Take-Off and Landing (PVTOL) robot navigating through obstacles to a goal region under stochastic noise.
- Setup: A safety-aware controller (TVLQR) was designed to ensure the closed-loop system is strongly contracting.
- Safety Guarantee: The system was required to satisfy a 99.99% safety guarantee ( $\delta = 10^{-4}$ ).
- Comparison: The probabilistic tube radius calculated using the proposed method was 0.54. In contrast, the bound derived from standard ISA was so large ( $>10$ ) that it was unvisualizable, rendering it useless for practical safety verification.
- Simulation: 10,000 stochastic trajectories were simulated. All trajectories remained collision-free, validating the theoretical safety guarantees.

5. Significance

This work provides a rigorous mathematical framework for safety-critical control in stochastic environments. By establishing tight concentration inequalities, it enables the design of controllers that can guarantee safety with high probability (e.g., $>99.9\%$ ) even in the presence of significant noise. This is crucial for applications such as autonomous driving, robotics, and power grids, where traditional conservative bounds often lead to overly restrictive or infeasible control policies. The ability to handle time-varying contraction makes the approach applicable to a broader class of dynamic systems than previous methods.