⚡ electrical engineering

Set-Based Value Function Characterization and Neural Approximation of Stabilization Domains for Input-Constrained Discrete-Time Systems

This paper proposes a novel framework for estimating domains of stabilization in input-constrained discrete-time systems by characterizing them through set-based value functions and employing a physics-informed neural network to learn these functions and synthesize stabilizing controllers.

Original authors: Mohamed Serry, S. Sivaranjani, Jun Liu

Published 2026-04-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mohamed Serry, S. Sivaranjani, Jun Liu

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 park a very tricky, self-driving car in a crowded garage. The car has a mind of its own (it's a nonlinear system), and the steering wheel has limits on how far you can turn it (input constraints).

Your goal is twofold:

The Safe Zone: Figure out exactly which starting spots in the parking lot allow the car to eventually park itself perfectly in the center without crashing or spinning out. In the paper, this is called the Domain of Stabilization (DOS).
The Driver: Create a set of instructions (a controller) that tells the car how to steer from any spot in that safe zone to the center.

The Old Way: Guessing and Checking

Traditionally, engineers tried to solve this by drawing a simple shape (like a circle or an oval) around the parking spot and saying, "If you start inside this circle, you're safe."

The Problem: This is often too conservative. It might say, "You can't park from this spot," even though a skilled driver could actually make it work. It's like saying you can't drive to the store unless you live within one block, ignoring that you could drive two blocks if you took a different route.
The Math: Old methods used "Lyapunov functions," which are like mathematical energy meters. If the energy goes down, you're safe. But finding the perfect energy meter for a complex car is incredibly hard and often leads to rough, inaccurate maps.

The New Way: The "Set-Based" Map and the "Smart Student"

This paper proposes a smarter, more flexible way to map the safe zone and teach the car how to drive.

1. The "Set-Based" Value Function (The Infinite Horizon Map)

Instead of looking at just one point on the map, the authors imagine looking at groups of points (sets) all at once.

The Analogy: Imagine you are a weather forecaster. Instead of asking, "Will it rain at this specific corner at 5 PM?", you ask, "If I drop a whole cloud of water droplets here, where will they end up in 10 minutes?"
The Innovation: They created a new type of "scorecard" (Value Function) that tracks how long it takes for a whole group of starting positions to reach the target. If the score is finite, you are in the safe zone. If it's infinite, you are doomed to spin out.
The Rulebook: They proved that this scorecard follows a specific rule (a Bellman-type equation). Think of this like a game rule: "Your score today equals the cost of this move plus the score of where you land next."

2. The "Physics-Informed" Neural Network (The Smart Student)

Now, how do we calculate this scorecard for a complex car? We can't solve the math equations by hand; they are too hard. So, they use an Artificial Neural Network (AI).

The Old AI: Usually, you train an AI by showing it thousands of examples of "good driving" and "bad driving." It learns by trial and error.
The New AI (Physics-Informed): This is the paper's secret sauce. Instead of just showing examples, they teach the AI the laws of physics directly.
- Imagine teaching a student to play chess. Instead of just showing them games they won, you give them the rulebook and say, "You must follow these rules in every move you make."
- The AI is trained to minimize the "error" in following the Bellman rulebook. It learns to predict the scorecard by ensuring its predictions obey the fundamental laws of the system's movement.

3. The Result: A Much Bigger Safe Zone

By using this method, the authors were able to draw a much more accurate map of the safe zone.

The Outcome: In their computer simulations (the "numerical examples"), the old methods drew a small, safe circle. The new method drew a large, irregular shape that hugged the actual limits of what the car could do.
The Driver: Once the AI learned the scorecard, it could instantly tell the car, "From this spot, turn left; from that spot, turn right," ensuring the car reaches the center safely.

Summary in a Nutshell

The Problem: It's hard to know exactly how far you can push a complex machine before it breaks, especially when you have limited control over it.
The Solution: The authors created a new mathematical "scorecard" that tracks groups of possibilities rather than single points.
The Tool: They taught an AI to learn this scorecard not just by memorizing data, but by strictly following the mathematical laws of the system (Physics-Informed Learning).
The Benefit: This gives us a much larger, more accurate "safe zone" and a better set of instructions to keep the system stable, which is crucial for things like autonomous vehicles, power grids, and robotic arms.

In short, they built a smarter, more honest map for the "safe zone" of complex machines, using an AI that learned the rules of the game rather than just guessing.

1. Problem Statement

The paper addresses the challenge of estimating the Domain of Stabilization (DOS) for nonlinear, discrete-time systems subject to input constraints.

Context: While many control methods (e.g., Model Predictive Control, Control Lyapunov Functions) exist, they often provide only local stability guarantees. The resulting domain of attraction (DOA) is frequently smaller than the true DOS, especially when input constraints are present.
The Gap: Existing value-function-based approaches (which theoretically characterize the exact DOS) are computationally intractable for general nonlinear systems due to the "curse of dimensionality" and the difficulty of solving infinite-horizon optimal control problems involving infimum operators over control inputs.
Objective: To develop a framework that accurately estimates the maximal DOS and synthesizes stabilizing controllers for input-constrained systems without relying on exhaustive state-space discretization.

2. Methodology

The authors propose a novel framework integrating set-based value functions with Physics-Informed Neural Networks (PINNs).

A. Set-Based Value Function Characterization

Instead of defining value functions on the state space $\mathbb{R}^n$ , the authors define them on the metric space of compact sets $\mathcal{K}(\mathbb{R}^n)$ equipped with the Hausdorff distance.

Definitions:
- Let $A$ be a Controlled Invariant Set (CIS).
- A cost function $\Psi(X)$ is defined as the infimum of a distance-based penalty over a set $X$ : $\Psi(X) = \inf_{y \in X} \alpha(y)$ .
- The Value Function $V$ is the infinite sum of costs over reachable sets: $V(X) = \sum_{k=0}^{\infty} \Psi(\mathcal{R}(X, k))$ , where $\mathcal{R}(X, k)$ is the reachable set at time $k$ .
- A transformed function $W(X) = 1 - \exp(-V(X))$ is introduced to map the infinite sum to a bounded range $[0, 1)$ .
Key Theoretical Result: Theorem 2 proves that the DOS ( $D_A$ ) is exactly the set of initial states $x$ for which $V(\{x\}) < \infty$ (or equivalently $W(\{x\}) < 1$ ). Thus, the sublevel sets of $W$ characterize the DOS.

B. Bellman-Type Functional Equations

The paper derives functional equations that these value functions satisfy, avoiding explicit infimum operators over controls in the loss function:

For $V$ : $v(X) = \Psi(X) + v(F(X))$ , where $F(X)$ is the image of set $X$ under the system dynamics and input constraints.
For $W$ : A Zubov-type equation is derived: $w(X) - w(F(X)) = \xi(X)(1 - w(F(X)))$ , where $\xi(X)$ depends on the cost $\Psi(X)$ .
Significance: These equations allow the value function to be learned by satisfying a residual error condition, similar to how PINNs solve PDEs.

C. Physics-Informed Neural Network (PINN) Framework

To approximate these set-based functions, the authors employ a learning framework:

Set Embedding: Since neural networks operate on vectors, the authors assume an injective mapping $T$ that embeds singleton sets $\{x\}$ and their reachable images $F(\{x\})$ into a finite-dimensional space $\mathbb{R}^L$ (e.g., using hyper-rectangles or zonotopes).
Loss Function: The training minimizes a composite loss $L(\theta) = \lambda_d L_d + \lambda_{pi} L_{pi}$ $L (θ) = λ_{d} L_{d} + λ_{p i} L_{p i}$ :
- Data-Driven Loss ( $L_d$ ): Minimizes the difference between the network output and a finite-horizon approximation of $W$ (computed via sampled trajectories).
- Physics-Informed Loss ( $L_{pi}$ ): Enforces the Bellman-type equation (9) by minimizing the residual: $J_{pi} = (W_{\theta}(X) - W_{\theta}(F(X)) - \xi(X)(1 - W_{\theta}(F(X))))^2$ .
Controller Synthesis: Once the network $W_{\theta}$ is trained, a stabilizing controller is synthesized by selecting control inputs that decrease the value function (i.e., $u$ such that $W_{\theta}(f(x, u)) < W_{\theta}(x)$ ).

3. Key Contributions

Set-Based Formulation: Introduction of value functions defined on metric spaces of compact sets, which characterizes the DOS exactly without requiring state-space discretization.
Novel Functional Equations: Derivation of Bellman-type equations for set-valued dynamics that eliminate the need for explicit infimum operators over control inputs, making them suitable for gradient-based learning.
PINN Integration: Development of a physics-informed learning framework that embeds the derived functional equations directly into the neural network training, ensuring the learned function respects the system dynamics and stability properties.
Constructive Control Synthesis: A method to extract a stabilizing feedback controller directly from the learned value function, guaranteeing asymptotic stability within the estimated domain.

4. Numerical Results

The framework was validated on two discrete-time nonlinear systems with input constraints:

System 1: A 2D nonlinear system.
System 2: A 3D nonlinear system.
Comparison: The neural network estimates of the DOS were compared against traditional quadratic Lyapunov function estimates.
Findings:
- The proposed method yielded significantly less conservative (larger) estimates of the DOS compared to quadratic Lyapunov approaches.
- The learned value function $W_{\theta}$ closely approached the theoretical limit of 1 near the boundary of the true DOS.
- The synthesized controllers successfully stabilized trajectories from initial conditions within the estimated domain while satisfying input constraints.
- Computational efficiency was demonstrated, with training times of ~5 minutes (2D) and ~15 minutes (3D) on a standard laptop.

5. Significance and Future Work

Theoretical Impact: This work bridges the gap between the rigorous theoretical guarantees of value-function-based control (which are often computationally impossible) and the scalability of modern deep learning. It provides a principled way to learn maximal Lyapunov functions for constrained systems.
Practical Impact: It offers a data-efficient, model-based approach to control synthesis that does not require massive datasets (unlike purely data-driven RL) and avoids the "curse of dimensionality" associated with grid-based dynamic programming.
Future Directions: The authors plan to integrate formal verification tools (e.g., $\alpha,\beta$ -CROWN, SMT solvers) to provide certified guarantees for the learned controllers and extend the framework to systems with state constraints and disturbances.

In summary, the paper presents a robust, mathematically grounded framework that leverages set-valued analysis and neural networks to solve the long-standing problem of estimating and controlling the stabilization domains of constrained nonlinear systems.