⚡ electrical engineering

Discontinuous integro-differential equations and sliding mode control

This paper extends the Filippov method and equivalent control approach to analyze and design sliding mode control systems governed by nonlinear discontinuous integro-differential equations and infinite-dimensional differential equations in Banach spaces, with specific applications to distributed input delay and heat control systems.

Original authors: Andrey Polyakov

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrey Polyakov

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: Taming the "Jittery" System

Imagine you are trying to steer a very heavy, slow-moving ship through a storm. You have a rudder (the control), but the ocean is full of unpredictable waves (disturbances).

Sliding Mode Control (SMC) is like a super-aggressive captain who doesn't just gently turn the rudder; they slam it hard to the left or right instantly to keep the ship on a straight line. It's a "bang-bang" approach: if you drift left, you jerk right; if you drift right, you jerk left.

The problem? In the real world, you can't actually move a rudder instantly from full-left to full-right. It takes time. Also, mathematically, this "instant switching" creates a problem: the equations describing the ship's movement break down. They become "ill-posed," meaning the math says the ship's future is undefined or impossible to calculate at the exact moment of the switch.

This paper is about fixing the math so we can design these aggressive, robust controllers for complex systems that have "memory" or "delays."

1. The Problem: The "Frozen" Moment

In standard physics, if you know where something is and how fast it's going, you can predict where it will be next. But with Sliding Mode Control, the controller switches so fast that at the exact moment of the switch, the rules of the game change instantly.

The Analogy: Imagine driving a car where the steering wheel snaps instantly from "Left" to "Right" every time you cross the center line. At the exact moment you are on the line, is the car turning left or right? The math gets confused. It's like a traffic light that is stuck halfway between Red and Green.

2. The Old Fix: Filippov's "Fuzzy" Box

Decades ago, a mathematician named Filippov came up with a clever trick to fix this. Instead of asking "Is the steering wheel Left or Right?", he said, "Let's assume the steering wheel is both Left and Right at the same time, effectively averaging them out."

The Analogy: Think of a spinning coin. When it's spinning, is it Heads or Tails? It's neither, but it's a mix of both. Filippov's method creates a "Fuzzy Box" (a mathematical set) that contains all possible outcomes. This turns the broken, "ill-posed" equation into a "well-posed" one where the math works again.

3. The New Challenge: Systems with "Memory"

The paper argues that most previous math only worked for simple systems (like a car moving on a flat road). But many real-world systems are more complex:

Distributed Delays: Imagine a long pipeline. If you turn a valve at one end, it takes time for the pressure change to reach the other end. The system "remembers" what you did in the past.
Heat Systems: If you heat one side of a metal rod, the heat takes time to travel to the other side.

These systems are modeled by Integro-Differential Equations (IDEs).

The Analogy: A normal equation is like a snapshot. An IDE is like a video recording. To know what happens now, you need to know the snapshot plus the entire history of the video up to this point. The "memory" (the integral part) makes the math much harder.

4. The Paper's Solution: Extending the "Fuzzy Box"

Andrey Polyakov (the author) asks: Can we use Filippov's "Fuzzy Box" trick for these complex systems with memory?

The Answer is Yes.

He extends the method to handle these "memory" systems. Here is how he does it, broken down:

A. Regularizing the Chaos

He proves that even if the control signal is switching wildly (like the snapping rudder), we can still define a valid path for the system.

The Metaphor: Imagine a chaotic dance floor where people are jumping randomly. The author creates a "dance floor boundary" (the mathematical inclusion) that contains all the possible jumps. Even though the dancers are jumping wildly, we can now predict the average flow of the crowd.

B. The "Equivalent Control" (The Secret Sauce)

Once the system is "regularized" (fixed), the author uses a tool called Equivalent Control.

The Analogy: When the ship is perfectly on course (the "sliding mode"), the captain isn't actually slamming the rudder left and right anymore. Instead, the rudder is holding a steady, invisible angle that perfectly cancels out the waves.
The Innovation: For systems with memory, this "invisible angle" isn't just a simple number; it's a solution to a complex equation that looks back at the past. The author shows how to calculate this "perfect steady angle" even when the system has delays.

5. Real-World Applications in the Paper

The author tests this theory on two specific examples:

Systems with Delayed Inputs:
- Scenario: A control signal takes time to travel through a network.
- Result: The author designs a controller that forces the system to ignore the delay and stabilize quickly, proving the math works even with the lag.
The Heat Equation (The Metal Rod):
- Scenario: You want to control the temperature of a metal rod. You can only measure the average temperature (output) and apply heat at one spot (input).
- The Twist: The rod is an "infinite-dimensional" system (it has infinite points of temperature). Usually, this is a nightmare for math.
- The Trick: The author realizes that even though the rod is infinite, the relationship between the input (heat) and the output (average temp) can be described by a simple, finite equation with memory (an IDE).
- Outcome: By using his new math, he designs a controller that forces the average temperature to zero (or a target) in a finite time, despite the heat traveling slowly through the metal.

Summary: Why Does This Matter?

Before: Engineers could design robust controllers for simple systems, but when they added "delays" or "memory" (like in heat, fluids, or networks), the math broke, and they had to guess or use approximations.
Now: This paper provides a rigorous mathematical "rulebook" for these complex systems. It tells engineers exactly how to design controllers that are:
1. Robust: They work even if there are unknown disturbances (like wind or noise).
2. Precise: They can handle systems that "remember" the past.
3. Safe: The math proves the system won't go crazy or become undefined.

In a nutshell: The author took a powerful tool for fixing broken math (Filippov's method) and upgraded it to handle systems that have a "memory," allowing us to build better, more reliable controllers for everything from heating systems to communication networks.

1. Problem Statement

The paper addresses the analysis and design of Sliding Mode Control (SMC) systems modeled by integro-differential equations (IDEs).

The Core Issue: Standard SMC theory relies on discontinuous feedback laws to reject matched disturbances. For finite-dimensional systems, these are modeled by discontinuous Ordinary Differential Equations (ODEs), which are mathematically ill-posed. The Filippov method is the standard tool to regularize these ODEs into well-posed Differential Inclusions (ODIs).
The Gap: The classical Filippov method is inapplicable to infinite-dimensional systems (e.g., those modeled by Partial Differential Equations - PDEs) or systems with delays because abstract spaces lack the notion of a "set of Lebesgue measure zero" required for the standard regularization.
The Specific Challenge: Many infinite-dimensional systems (like heat equations) and systems with distributed delays can be reduced to finite-dimensional IDEs describing their input-output dynamics. However, the theory for regularizing discontinuous IDEs and applying SMC design to them has not been developed.

2. Methodology

The author extends the classical theory of discontinuous dynamical systems to the class of IDEs.

A. Extension of the Filippov Method to IDEs

The paper defines a Filippov solution for the IDE:
$\dot{x}(t) = f(t, x(t)) + \int_{t_0}^t \Phi(t, \tau) \tilde{f}(\tau, x(\tau)) d\tau$
where $f$ and $\tilde{f}$ are discontinuous in $x$ .

Regularization: The discontinuous functions are replaced by set-valued mappings $K[f]$ and $K[\tilde{f}]$ (convex hulls of limits of the function values near the discontinuity).
Resulting Inclusion: The system is regularized into a differential inclusion:
$\dot{x}(t) \in K[f](t, x(t)) + \int_{t_0}^t \Phi(t, \tau) K[\tilde{f}](\tau, x(\tau)) d\tau$
Existence: Using the Arzelà–Ascoli theorem and properties of Aumann integrals, the author proves the existence of Filippov solutions for these IDEs under standard measurability and boundedness conditions.

B. Uniqueness Analysis

The paper establishes conditions for the uniqueness of Filippov solutions, which is non-trivial for IDEs due to the lack of the semigroup property (unlike ODEs).

Sliding Mode Case: Uniqueness is proven if the discontinuity set of the integral term ( $\tilde{f}$ ) does not intersect the discontinuity set of the differential term ( $f$ ), and specific tangency conditions are met.
Switching Case: Uniqueness is guaranteed if trajectories immediately leave the discontinuity set.

C. Equivalent Control Method for IDEs

The paper extends the Equivalent Control Method (a cornerstone of SMC design) to IDEs.

For affine-in-control IDEs ($f = a + bu$, $\tilde{f} = \tilde{a} + \tilde{b}u$ ), the equivalent control $u_{eq}$ is derived.
Unlike ODEs where $u_{eq}$ is found via an algebraic equation, for IDEs, $u_{eq}$ is the solution to a Volterra integral equation of the second kind.
The paper proves that for single-input systems, the set of Utkin solutions (based on equivalent control) coincides with Filippov solutions.

3. Key Contributions

Theoretical Framework: First extension of the Filippov regularization and equivalent control methods to finite-dimensional discontinuous IDEs.
Well-Posedness: Proofs of existence, uniqueness, and continuous dependence on initial data for Filippov solutions of IDEs.
SMC Design for Delay Systems: Development of a sliding mode controller for systems with distributed input delays, modeled as IDEs. The control gain is derived to ensure finite-time convergence to the sliding surface.
Infinite-Dimensional Reduction: A novel approach to SMC for Banach space systems (e.g., PDEs). The author demonstrates that the input-output dynamics of such systems can be represented as finite-dimensional IDEs. This allows the application of finite-dimensional SMC theory to infinite-dimensional problems without dealing directly with the ill-posedness of the PDE.
Numerical Validation: Implementation of explicit Euler methods for IDEs and simulation of a heat control system to validate the theoretical predictions.

4. Key Results

Finite-Time Stability: For a linear IDE with distributed delay, a discontinuous feedback law $u(t) = -\rho (CB)^{-1} \frac{y(t)}{\|y(t)\|}$ is shown to drive the output $y(t)$ to zero in finite time, provided the control gain $\rho$ satisfies specific bounds involving the delay kernel and disturbance bounds.
Sliding Motion Dynamics: Once on the sliding surface, the system dynamics are governed by a reduced-order equation involving the equivalent control, which is explicitly defined via a convergent Neumann series solution to the Volterra integral equation.
Heat System Application:
- A heat equation with boundary control and unknown disturbances is modeled.
- The input-output dynamics are derived as an IDE.
- A discontinuous control law is designed to force the output (a weighted integral of the temperature) to zero in finite time.
- Stability: The closed-loop system is proven to be globally exponentially stable in the sliding mode.
- Simulation: Numerical simulations confirm that the output converges to zero and the control signal exhibits the expected chattering (approximated by the numerical method), validating the theoretical model.

5. Significance

Bridging Finite and Infinite Dimensions: The paper provides a rigorous mathematical bridge between finite-dimensional control theory (ODEs/IDEs) and infinite-dimensional systems (PDEs). It shows that for many PDEs with finite-dimensional inputs/outputs, one can bypass complex infinite-dimensional regularization by working with the equivalent finite-dimensional IDE.
Robustness: The methodology allows for the design of robust controllers for systems with distributed delays and infinite-dimensional dynamics, which are notoriously difficult to control using standard SMC techniques due to well-posedness issues.
Practical Applicability: By defining the equivalent control as a solution to a Volterra equation, the paper offers a constructive method for analyzing the "sliding mode" behavior of complex systems, moving beyond abstract existence proofs to calculable dynamics.
Future Directions: The work lays the foundation for extending other advanced SMC techniques (e.g., High-Order Sliding Mode, Implicit Discretization) to IDEs and Banach space systems.

In summary, Polyakov successfully generalizes the fundamental tools of Sliding Mode Control to a broader class of systems (IDEs), enabling the rigorous analysis and design of robust controllers for systems with delays and distributed parameter systems (PDEs).