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Estimation of the input-to-state stability gain functions from finite-dimensional approximations

This paper establishes a rigorous connection between the input-to-state stability of infinite-dimensional linear boundary control systems governed by analytic semigroups and their finite-dimensional approximations, demonstrating that ISS gain functions can be reliably computed from these numerical schemes.

Original authors: Birgit Hillebrecht, Benjamin Unger

Published 2026-03-03

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Birgit Hillebrecht, Benjamin Unger

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

The Big Picture: Predicting the Future of a Wobbly System

Imagine you are trying to predict how a cup of coffee will cool down. If the room is still, you can easily guess the temperature. But what if someone keeps opening the door, letting in cold drafts (the "input")? The coffee's temperature (the "state") will fluctuate based on how much cold air comes in.

In engineering and physics, we call this Input-to-State Stability (ISS). It's a way of saying: "If the outside disturbances (inputs) are small, the system's reaction (state) will stay small and eventually settle down."

For simple systems (like a few connected springs), mathematicians have known how to calculate exactly how much the system will wobble for a long time. But for complex systems described by partial differential equations (like heat spreading through a metal plate, or fluid flowing in a pipe), the math gets incredibly messy, especially when the "disturbance" happens right at the edge (the boundary).

The Problem:
We have powerful computers that can simulate these complex systems by breaking them down into tiny, simple pieces (finite-dimensional approximations). However, until now, there was no rigorous proof that the "wobble limits" calculated by the computer actually match the real, infinite complexity of the physical world. It was like trusting a weather app's forecast without knowing if the math behind it was actually correct for a hurricane.

The Solution:
This paper provides a rigorous "bridge." It proves that if you simulate a complex system on a computer using a specific method, the stability numbers you get from the simulation do converge to the true, real-world numbers as you make the simulation finer and finer.

Key Concepts Explained with Analogies

1. The Infinite vs. The Finite (The Ocean vs. The Grid)

The Real World (Infinite-Dimensional): Imagine the ocean. It has infinite points of water. You can't count them all. This is the "infinite-dimensional system."
The Computer Model (Finite-Dimensional): To simulate the ocean, a computer puts a grid over it. It only calculates the water level at specific grid points. This is the "finite-dimensional approximation."
The Challenge: The paper asks: "If I make the grid smaller and smaller (more points), will my calculation of the ocean's stability eventually match the real ocean?"

2. The "Unbounded" Operator (The Edge Case)

In many physics problems, the disturbance happens at the boundary (the edge of the domain).

Analogy: Imagine a guitar string. If you pluck the middle, the math is easy. But if you shake the end of the string (the boundary) violently, the math gets weird. The force applied at the edge is "unbounded" relative to the rest of the string.
The Paper's Trick: The authors use a mathematical sleight-of-hand called the Fattorini Trick.
- Imagine: You want to measure the effect of shaking the end of the string, but your calculator only understands forces applied in the middle.
- The Trick: You mathematically "translate" the shaking at the edge into an equivalent "ghost force" inside the string. This allows the computer to handle the boundary problem using standard tools.

3. The "Pre-Boundary Closure" (The Blueprint)

This is the paper's most creative contribution.

The Analogy: Imagine you are building a model of a house.
- System A (The Real House): Has walls, a roof, and a front door where people enter.
- System D (The Computer Model): Usually, we just model the rooms inside and ignore the door, forcing the door to be closed.
- System C (The Pre-Boundary Closure): The authors suggest building a "skeleton" model first. This model includes the door and the frame, but the door is open.
Why it matters: By modeling the "skeleton" (System C) first, they can mathematically prove that when they close the door and turn it into the standard computer model (System D), the stability properties remain consistent with the real house (System A). It's a safety net that ensures the translation from "real world" to "computer code" doesn't lose any critical information.

4. The "Gain Function" (The Safety Margin)

The ultimate goal is to find a Gain Function (denoted as $\gamma$ ).

Analogy: Think of this as a "Safety Margin" or a "Speed Limit."
- If the wind (input) blows at 10 mph, the tree (system) might sway 2 feet.
- If the wind blows at 20 mph, the tree might sway 4 feet.
- The Gain Function is the formula that tells you: "No matter how hard the wind blows, the tree will never sway more than X feet."
The Paper's Achievement: They show that you can calculate this "Safety Margin" by running the computer simulation. As the simulation gets more detailed, the calculated safety margin gets closer and closer to the true, perfect safety margin.

Why Does This Matter? (The "So What?")

Trust in AI and Neural Networks: The authors mention that this math is crucial for Physics-Informed Neural Networks (PINNs). These are AI models that try to learn physics. If an AI makes a mistake at the boundary (like predicting the wrong temperature at the edge of a metal plate), we need to know exactly how much that error will mess up the entire prediction. This paper gives us the tool to calculate that error bound.
No More "Black Box" Guessing: Previously, engineers had to guess if their computer simulations were stable enough. Now, they have a rigorous mathematical proof that if they follow these steps, the computer's answer is trustworthy.
Handling the "Hard" Stuff: It specifically solves the problem of systems where the control happens at the edge (boundaries), which is the most common and difficult scenario in real-world engineering (like controlling the temperature of a furnace or the pressure in a pipeline).

Summary in One Sentence

This paper builds a mathematical "translation bridge" that proves we can safely use simple computer simulations to calculate the exact safety limits of complex, real-world systems, even when the disturbances happen at the very edges of those systems.

1. Problem Statement

The paper addresses a critical gap in control theory: the rigorous connection between Input-to-State Stability (ISS) of infinite-dimensional systems with unbounded control operators and their finite-dimensional approximations.

Context: ISS is a fundamental concept for analyzing system stability in the presence of external inputs. While well-established for finite-dimensional systems and bounded operators, extending ISS analysis to infinite-dimensional systems (specifically Partial Differential Equations or PDEs) with boundary control is challenging.
The Challenge: In PDEs with boundary control (e.g., heat equation with Dirichlet boundary conditions), the control operator $B$ is typically unbounded. Transforming such systems into the abstract form $\dot{x} = Ax + Bu$ requires the use of extrapolation spaces ( $X_{-1}$ ), making $B$ a bounded operator from the input space $U$ to $X_{-1}$ .
The Gap: Existing numerical techniques often fail to provide rigorous bounds on the ISS gain functions ( $\beta$ and $\gamma$ ) for these unbounded cases. Specifically, it is unclear how to compute the gain $\gamma$ (which bounds the state response to the input) using standard finite-dimensional discretizations (like Finite Element or Finite Difference methods) without relying on explicit analytical knowledge of the operator's spectrum or resolvent, which is often unavailable for complex geometries.
Motivation: The authors aim to provide computable ISS gains to support error estimation for Physics-Informed Neural Networks (PINNs), where boundary mismatches act as inputs. Current error bounds require differentiable inputs or simple geometries; this work seeks to remove those restrictions.

2. Methodology

The authors develop a framework based on semigroup theory and Trotter-Kato approximation theory to bridge the infinite-dimensional and finite-dimensional worlds.

A. Mathematical Framework

System Class: The study focuses on linear boundary control systems where the autonomous evolution is governed by an analytic semigroup (typical of parabolic PDEs).
Extrapolation Spaces: They utilize the extrapolation space $X_{-1}$ to handle unbounded operators. The control operator $B$ maps $U \to X_{-1}$ .
Fractional Powers: A key technical tool is the use of fractional power spaces $X_{\alpha}$ (specifically $X_{-1+\alpha}$ ). By leveraging the analyticity of the semigroup, the authors bound the system response using the norm of the operator $(-A)^{-1+\alpha}B$ .
Approximation Sequence: They consider a sequence of finite-dimensional spaces $X_n$ with projection operators $P_n$ and extension operators $E_n$ . The discrete system is $\dot{x}_n = A_n x_n + B_n u$ .

B. The "Pre-Boundary Closure" Strategy

To handle boundary control systems (where $B$ is not explicitly given but derived from a boundary operator $D$ ), the authors introduce a novel intermediate step:

Fattorini Trick: They recall that a boundary system $\dot{x}=Ax, Dx=u$ can be transformed into a distributed system $\dot{x}=Ax+Bu$ where $B = AD_0 - A^{-1}D_0$ ( $D_0$ is a right inverse of $D$ ).
Discrete Intermediate System: Instead of discretizing the distributed form directly, they propose a pre-boundary closure discretization. This involves discretizing the boundary operator $D$ and the domain simultaneously to create an intermediate system that preserves the boundary structure.
Convergence Proof: They prove that if the discrete boundary operators converge appropriately, the resulting discrete control operators $B_n$ converge to the true unbounded operator $B$ in the strong operator topology of the fractional power space $X_{-1+\alpha}$ .

3. Key Contributions

The paper makes three primary theoretical contributions:

ISS Gain Estimation for Bounded Operators (Theorem 3.7):
For abstract systems where $B$ is known and bounded into $X_{-1+\alpha}$ , the authors prove that the ISS gains $\beta$ and $\gamma$ can be computed as limits of the gains derived from finite-dimensional approximations. They provide explicit formulas for these gains based on the convergence of the semigroup growth bounds ( $M_n, \omega_n$ ) and the resolvent bounds.
Extension to Boundary Control Systems (Theorem 4.1):
This is the core innovation. The authors extend the results to cases where $B$ is not explicitly available but arises from a boundary condition. They introduce the pre-boundary closure discretization and prove that under specific consistency assumptions (convergence of right inverses, uniform boundedness of discrete operators), the discrete control operators $B_n$ converge to the true $B$ in the required norm. This allows the computation of ISS gains for boundary control systems without explicit knowledge of the extrapolated operator $A^{-1}$ .
Removal of Differentiability Requirements:
Unlike previous works (e.g., [10]) that required the input $u$ to be time-differentiable to derive error bounds, this framework restores the standard $L^\infty$ -ISS estimate, allowing for non-smooth inputs. This significantly broadens the applicability of the theory to practical engineering problems.

4. Results and Numerical Validation

Theoretical Bounds: The paper derives computable constants $K_1$ and $K_2$ (dependent on the semigroup type and resolvent bounds) that define the ISS gain $\gamma$ .
Numerical Example: The authors apply their theory to the 1D Heat Equation with Dirichlet boundary control.
- They use a standard finite-difference discretization.
- They verify all assumptions (uniform sectoriality, resolvent convergence, boundedness of discrete boundary inverses).
- Outcome: The numerically computed ISS gain $\gamma(s) \approx 0.8989 s$ closely matches the rigorous analytical result for a similar problem found in literature ( $\gamma(s) \approx 0.8165 s$ ).
- Convergence: The paper demonstrates the convergence of the discrete parameters (growth bound $\omega_n$ , resolvent bound $D_n$ , and operator norm $\|(-A_n)^\alpha B_n\|$ ) as the mesh size $n$ increases.

5. Significance and Impact

Bridging Theory and Computation: The work provides the first rigorous framework to compute ISS gains for infinite-dimensional systems with unbounded boundary controls using standard numerical discretizations.
Enabling PINN Error Bounds: By providing computable ISS gains for complex geometries and non-smooth inputs, the paper enables the practical use of a posteriori error estimators for Physics-Informed Neural Networks, a rapidly growing field in scientific machine learning.
Systematic Discretization Strategy: The introduction of the "pre-boundary closure discretization" offers a systematic way to switch between boundary trace formulations and abstract Cauchy problems, which is valuable for other operator-theoretic applications beyond ISS.
Future Directions: The authors identify limitations (e.g., applicability to hyperbolic systems or non-linear systems) and suggest future work combining this framework with Riccati-based approximation methods.

In summary, this paper successfully establishes a rigorous mathematical bridge allowing engineers and mathematicians to estimate stability properties of complex PDE control systems using finite-dimensional numerical simulations, even when the control acts on the boundary.