Stabilization of Unidirectional First-Order PDE-ODE Coupled Systems with Boundary and Distributed Input Delays

arXiv:2610.12226 · eess.SY, cs.SY, math.AP, math.OC · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Stabilization of Unidirectional First-Order PDE-ODE Coupled Systems with Boundary and Distributed Input Delays".

Dev: The gist The proposed method can simultaneously address distinct delays of boundary control and distributed control without requiring ordering of the delays <ref:2610.12226#pg6>.

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: So we’re going into how this paper tackles this challenge, focusing on the title itself. Rosa, what's your take on what "Stabilization of Unidirectional First-Order PDE-ODE Coupled Systems with Boundary and Distributed Input Delays" actually implies for us right now?

Dev: It implies they are solving a problem that used to require you to impose a strict order on the delays before you could even think about stabilizing the system. They're showing that you don't need that ordering constraint.

Taro: That sounds like it opens up a lot of possibilities for real-world applications where control inputs happen at different physical locations, maybe in a large industrial setup. It suggests a more flexible way to design controllers when the physical layout dictates heterogeneous delays.

Rosa: Right. The authors are using this approach to simultaneously handle distinct boundary and distributed control inputs without having to worry about one delay always being smaller than the other, which is a major simplification if you're designing something practical.

Dev: They achieve this by designing a set of three backstepping transformations that involve both Volterra-type and Fredholm-type integral operators. That’s the main tool they build to handle those mixed delay types.

The paper's summary: Rosa: So, putting that together, what does the actual core mechanism of this paper look like? What are they doing fundamentally to get from an unstable system to a stable one?

Dev: They are using these three specific backstepping transformations. Two of those transformations are designed specifically to compensate for the boundary delay, and the third one is built to handle that distributed delay, which is where it gets more complicated.

Taro: When you talk about transforming systems like this with integral operators, I wonder how they manage the complexity introduced by those kernels? Are we talking about simple algebra or something more involved when you're dealing with five PDE-associated and three ODE-associated kernel functions?

Rosa: It’s intricate. They define these kernels on rectangular domains and their governing equations have to satisfy two specific boundary conditions, which adds another layer of constraint to the whole transformation process.

Dev: To make sure this complex setup is actually solvable, they proved the well-posedness of those kernel equations using a rigorous method involving the method of characteristics and successive approximations. That’s how they ensure their mathematical machinery holds up.

The paper's improvements: Rosa: Okay, moving past the math, what are some of these actual practical improvements they propose? What does this mean for someone who is actually trying to build a controller for a physical system?

Dev: The main improvement they highlight is their ability to address the heterogeneous delays—the boundary delay versus the distributed delay—simultaneously without requiring any specific ordering assumptions between them. That’s huge because real systems rarely have delays that are neatly ordered like that.

Taro: So, if you can handle distinct delays at different locations, what does that allow us to do in terms of system modeling? Does it mean we can model more realistic industrial setups where actuators are physically separated in different ways?

Rosa: Precisely. They show how this method successfully maps the original delayed unstable system into a target system that is stable, and they prove exponential stability in the L infinity norm under this new controller. That's a strong guarantee for stability.

Dev: And they also managed to resolve some issues with the ill-posedness that could arise from boundary conditions being insufficient, by introducing a modified boundary condition that keeps things consistent across the rectangular domain. That’s another fix for real implementation problems.

Conclusion: Rosa: So we're wrapping up this discussion on "Stabilization of Unidirectional First-Order PDE-ODE Coupled Systems with Boundary and Distributed Input Delays." What's the big picture implication for people who only listen to the show?

Dev: Essentially, this paper gives us a robust method to design controllers for complex systems that have both boundary and distributed input delays, no matter how different those delays are. It translates an unstable system into a stable one using these specialized transformations.

Taro: For autonomy researchers like me, this suggests we can tackle more realistic scenarios in autonomous navigation or industrial robotics where the control inputs aren't all happening in one neat spot. We can design controllers that work even when the environment misbehaves and introduces unpredictable time lags.

Rosa: I think the key thing is that they achieve exponential stability in the L infinity norm under this delay-compensated controller, which means we have a very strong proof about how fast the system settles down after a disturbance. It’s a solid piece of control theory applied to these specific PDE-ODE systems.

Dev: And from an engineering standpoint, the numerical validation was pretty compelling. They tested it on an unstable linear system with a 2D ODE and even on a screw extrusion model for melt spinning, showing that the controllers actually stabilized the system where nominal controllers failed in the presence of delays <ref:2610.12226#pg2>.

Taro: That practical validation is what really sells it—seeing it work not just in theory, but when you plug in something like a screw extrusion model. It moves this from abstract math to something that has tangible results for process control.

Rosa: So, "Stabilization of Unidirectional First-Order PDE-ODE Coupled Systems with Boundary and Distributed Input Delays" gives us a way to handle those mixed delays flexibly, providing a mathematically sound method for getting stable closed-loop systems in these types of coupled transport models. That’s where we'll leave it for now.

Sanguan Zhong, Jie Qi

eess.SY, cs.SY, math.AP, math.OC

Submitted: 2026-10-08

Updated: 2026-10-08

Comments: 20 pages, 27 figures. Extended version of the journal paper containing complete technical proofs

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

The gist: The gist The proposed method can simultaneously address distinct delays of boundary control and distributed control without requiring ordering of the delays <ref:2610.12226#pg6>.

Key concepts

Unidirectional First-Order PDE-ODE System: This describes a mathematical model involving coupled partial differential equations (PDEs) and ordinary differential equations (ODEs) that model two-phase advective transport. These systems are used to simulate physical processes where mass and energy move through space over time, incorporating both spatial diffusion/advection and temporal dynamics.
Boundary Control Delay: This refers to a delay in the input signal applied specifically at the edges or boundaries of the spatial domain. The paper deals with situations where the control action at one physical edge of the system is received later than expected, complicating stability analysis.
Distributed Input Delay: This occurs when control signals are applied across a continuous region (a distributed input) rather than just at discrete points. The delay varies depending on the exact location within that region, making it more complex to handle in control design compared to simple boundary delays.

Terminology

Closed-Loop System — 9× in this paper · explained in 4 episodes
Ordinary Differential Equations (ODEs) — 3× in this paper · explained in 2 episodes
Robustness — 3× in this paper · explained in 5 episodes
Consistency — 2× in this paper · explained in 3 episodes
Lyapunov function — 2× in this paper · explained in 2 episodes
Bifurcation Analysis — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes

Summary

The gist The proposed method can simultaneously address distinct delays of boundary control and distributed control without requiring ordering of the delays <ref:2610.12226#pg6>.

System Modeling and Challenges

The paper considers a system of unidirectional first-order hyperbolic partial differential equations (PDEs) coupled with ordinary differential equations (ODEs), modeling two-phase advective transport processes subject to both boundary and distributed input delays. This system is defined over two spatial domains, involving coupled dynamics where the open-loop system would be unstable if the coefficients θ and di were sufficiently large. Time delays are inherent in such systems due to the finite propagation speeds of mass, energy, and information. The research focuses on a more general configuration where boundary and distributed actuators are physically distinct, leading to different delay values.

Backstepping Transformation Design

To address the challenge of heterogeneous delays, the authors propose a set of three backstepping transformations involving both Volterra- and Fredholm-type integral operators. These transformations involve five PDE-associated and three ODE associated kernel functions. Two transformations are designed to compensate for the boundary delay, whereas the third compensates the distributed delay, making its construction more intricate. The kernels arising from the Fredholm-type terms are defined on rectangular domains, and their governing first-order PDEs are subject to two boundary conditions.

Well-Posedness Analysis of Kernel Equations

The well-posedness of the kernel equations is established through a rigorous analysis using the method of characteristics and successive approximations. The proof involves analyzing three groups of coupled kernel equations, which are solved sequentially for the first group, followed by a recursive iteration for the second group. The well-posedness of the inverse kernel system is also proven through analyzing the operator T j, which is shown to be bounded based on explicit expressions for the kernels.

Stability of the Closed-Loop System

The closed-loop system is proven to admit a unique weak solution in C([0, ∞); S) and its zero equilibrium is exponentially stable in the L∞ norm under the delay-compensated controller. This stability is demonstrated by analyzing the explicit solution of the target system, where estimates like V1(t) ≤ Γ0 e −ν0tV1(0) are derived. Furthermore, the norm equivalence between the Lyapunov functions V1(t) and V2(t) is shown to hold, establishing isomorphism between the state spaces of the original and target systems.

Numerical Validation

The effectiveness of the proposed method is validated through two numerical examples. The first example involves an unstable linear system with a 2D ODE, and the second is a screw extrusion model for the melt spinning process, which is linearized around its equilibrium point. Simulation results illustrate that the delay-compensated controllers successfully stabilize the system, in contrast to the nominal controllers which fail to achieve stabilization in the presence of input delays. The results for the screw extrusion model show that all states converge to their desired values within 10 seconds under the proposed controller.

Conclusion

This paper presents a control design methodology to compensate for both boundary and distributed input delays occurring in a two-phase advective unidirectional first-order PDE-ODE system. The proposed method can be applied to distinct delays without any ordering constraints. The method successfully maps the delayed unstable system to a stable target system, resulting in exponential stability of the closed-loop system in the L∞ norm. The results show that the proposed controllers can stabilize the nonlinear system with a fast convergence rate.

How it works

The control laws are obtained by applying the transformations (4)–(6) to convert the original system into a target system. The final controllers U1(t) and U2(z, t) are derived from these transformations, incorporating the kernel functions and the inverse kernels. The control law for the distributed input delay is intricate, involving terms like Z t t−D k(z, t − ς D, y)U2(y, ς d s dy.

Key Contributions

The contributions of the paper are as follows:

The method proposed can simultaneously address distinct delays of boundary control and distributed control without requiring ordering of the delays, unlike existing methods that require specific delay ordering assumptions.
To resolve the ill-posedness arising from insufficient boundary conditions, which leads to non-uniqueness of the solution on part of the rectangular domain, we introduce a modified boundary condition that replaces the original one while maintaining consistency.
A unified backstepping transformation is proposed for cross-domain first-order hyperbolic systems with distributed input delay. By combining the Dirac delta function and the Heaviside function in the kernel function, the proposed transformation eliminates unstable terms while preserving a triangular target structure.

References

[1] U. J. F. Aarsnes, R. Vazquez, F. Di Meglio, and M. Krstic, “Delay robust control design of under-actuated PDE-ODE-PDE systems,” in Proc. Amer. Control Conf., 2019, pp. 3200–3205

[4] H. Anfinsen and O. M. Aamo, Adaptive control of hyperbolic PDEs <ref:

Improvements for AI systems

Improved system can stabilize complex industrial processes by compensating for both boundary and distributed input delays without requiring specific delay ordering assumptions, as The method proposed can simultaneously address distinct delays of boundary control and distributed control without requiring ordering of the delays.
Improved system can manage multi-region transport dynamics like melt-spinning screw extrusion more accurately by using a delay-compensated controller that achieves stabilization even when boundary and distributed actuators are physically distinct, as demonstrated by the simulation results where the delay-compensated controllers successfully stabilize the nonlinear extrusion system.
Improved system can handle heterogeneous control inputs in continuous processes by utilizing three backstepping transformations involving both Volterra- and Fredholm-type integral operators, which allows for the joint compensation of distinct boundary and distributed delays.