Keywords: Lyapunov-Based and Backstepping Techniques
Abstract: Lyapunov theory is one of the cornerstones of stability analysis for linear and nonlinear partial differential equations. It provides powerful tools for understanding the long-term behavior of solutions, including asymptotic stability of equilibria, properties of attractors, and characterizations of robustness. In many classical treatments — especially in the nonlinear setting — Lyapunov functions are required to satisfy a coercivity condition from the outset, meaning they control the full state norm, in order to conclude robust stability results. On the other hand, it is well known from the linear context that quadratic Lyapunov functions are a particularly versatile tool — and such functions are often not coercive. In fact, there is no theoretical necessity for coercivity in the linear case, raising the question of whether similar flexibility might be possible in the nonlinear infinite-dimensional setting.

Recent developments have demonstrated that coercivity is not always essential even for nonlinear PDEs. In particular, it is now understood that global asymptotic stability can, in many cases, be characterized using noncoercive Lyapunov functions, which relax the requirement of full norm control. This opens the door to new possibilities in the analysis of infinite-dimensional systems, where coercivity may be a too strong requirement or may fail altogether.

In this talk, we explore the use of noncoercive Lyapunov functions in the stability theory of PDEs. We will focus on what kinds of stability properties can still be inferred when coercivity is absent, with special attention to the concept of input-to-state stability — a framework for analyzing the robustness of systems under external disturbances. Through a concrete example of a linear PDE, we will illustrate how a stability proof can rely crucially on a noncoercive Lyapunov function. Indeed, for this particular example no counterpart of a Lyapunov based proof using coercivity is currently known.

Finally, we will discuss how these ideas extend to interconnected PDE systems, where classical approaches often rely on strong assumptions such as coercivity. In particular, we will highlight recent progress in adapting small-gain techniques—a key method for assessing the stability of coupled systems within the input-to-state stability framework—to settings where Lyapunov functions may be noncoercive. These developments point toward a more flexible and robust theory for stability analysis in distributed and networked PDE systems.

Keywords: Lyapunov-Based and Backstepping Techniques, Traffic Control and Network Congestion, Flow Control
Abstract: In this paper, we investigated the robust stabilization problem for Aw–Rascle–Zhang (ARZ) traffic systems considering stochastic traffic demand from the upstream boundary represented by a Markov-jumping process. We propose a control law that combines operator learning with the backstepping control method. To enhance computational efficiency, the backstepping kernels used in the control law are approximated by neural operators (NOs). We demonstrate that mean-square exponential stability of the closed-loop system, with a nominal neural operator-approximated backstepping control law, can be achieved through Lyapunov analysis. The theoretical results are validated by numerical simulations.

Keywords: Optimal Control, Computational Methods, Flow Control
Abstract: Platooning of connected and autonomous vehicles (CAVs) plays a vital role in modernizing highways. This paper explores the significance of platooning in smart highways, employing a coupled partial differential equation (PDE) and ordinary differential equation (ODE) model to elucidate the complex interaction between bulk traffic flow and CAV platoons. Inspired by Dyna-Q algorithm, our study focuses on developing a Dyna-style planning and learning framework tailored for platoon control, with a specific goal of reducing fuel consumption. By harnessing the coupled PDE-ODE model, we improve data efficiency in Dyna-style learning through virtual experiences. Simulation results validate the effectiveness of our macroscopic model in modeling platoons within mixed-autonomy settings, demonstrating a notable 10.11% reduction in vehicular fuel consumption compared to conventional approaches.

Keywords: Traffic Control and Network Congestion, Systems Engineering, Flow Control
Abstract: The uncertainty in human driving behaviors leads to stop-and-go instabilities in freeway traffic. The traffic dynamics are typically modeled by the Aw-Rascle-Zhang (ARZ) Partial Differential Equation (PDE) models, in which the relaxation time parameter is usually unknown or hard to calibrate. This paper proposes an adaptive boundary control design based on neural operators (NO) for the ARZ PDE systems. In adaptive control, solving the backstepping kernel PDEs online requires significant computational resources at each timestep to update estimates of the unknown system parameters. To address this, we employ DeepONet to efficiently map model parameters to kernel functions. Simulations show that DeepONet generates kernel solutions nearly two orders of magnitude faster than traditional solvers while maintaining a loss on the order of (10^{-2}). Lyapunov analysis further validates the stability of the system when using DeepONet-approximated kernels in the adaptive controller. This result suggests that neural operators can significantly accelerate the acquisition of adaptive controllers for traffic control.

Keywords: Traffic Control and Network Congestion, Stability Theory, Lyapunov-Based and Backstepping Techniques
Abstract: This paper investigates the boundary feedback control problem for the mixed-autonomy traffic flow system consisting of Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs). Considering different spacing strategies of AVs, the stochastic impact area of AVs is modeled by a continuous Markov chain. The macroscopic traffic dynamics are described by an extended stochastic ARZ traffic model with Markov jumping parameters, consisting of fourth-order Partial Differential Equations (PDEs). Based on the measurements of the speed and density of HVs and AVs at the boundary of the road segment, a boundary feedback control law, which is implemented by regulating the outlet Variable Speed Limits (VSLs) and a ramp metering at the inlet boundary, is designed to drive the traffic states of both HVs and AVs to desired states. Then sufficient conditions for ensuring the exponential mean-square stability of the system are developed in terms of Linear Matrix Inequalities (LMIs) by employing the Lyapunov function method. Finally, the effectiveness of the proposed boundary feedback controller is validated by numerical examples.

Keywords: Traffic Control and Network Congestion, Lyapunov-Based and Backstepping Techniques, Stability Theory
Abstract: Unlike finite-time stability, where the upper bound of the settling time depends on the initial states of the system, the settling time in fixed-time stability is independent of initial states, and thus it can be adjusted in advance. This paper considers the fixed-time boundary stabilization problem of the Aw-Rascle-Zhang (ARZ) traffic model, which is a 2 × 2 coupled hyperbolic system. To stabilize the traffic speed and density to desired steady states in a fixed time, a full state feedback boundary controller is proposed, the practical actuator of which is performed through ramp metering at the downstream boundary. By utilizing spatial transformation and backstepping method, the linearized ARZ model is transformed into a fixed-time stable target system. We employ the characteristics and Lyapunov methods to prove that the linearized ARZ model is fixed-time stable in the sense of L2-norm. Furthermore, the upper bound of the settling time is determined solely by the control parameters, and remains independent of initial states. Finally, the numerical simulation is given to validate the theoretical results.

Keywords: Traffic Control and Network Congestion
Abstract: We propose an event-triggered control (ETC) strategy for the Aw-Rascle-Zhang (ARZ) traffic model under congested conditions. The considered ARZ-type model, governed by first-order hyperbolic partial differential equations (PDEs), captures traffic dynamics involving both Adaptive Cruise Control (ACC) and human-driven vehicles. Control actions adjust the time gap for ACC vehicles and are updated based on a suitable triggering rule. We conduct the stability analysis on a linearized and transformed system (a 2 by 2 linear hyperbolic system with in-domain control), and we asses input-to-state stability (ISS) with respect to actuation errors. A small-gain approach guides the design of appropriate triggering rule (small-gain event based triggering condition), ensuring exponential stability while preventing the Zeno phenomenon. Numerical simulations demonstrate the effectiveness of the proposed control strategy in stabilizing traffic flow.

Keywords: Model Reduction for Control
Abstract: For the non-collocated output regulation for wave equations, this paper postulates that the measurable outputs encompass the tracking error and even its perturbed time derivative influenced by an unknown output disturbance. An adaptive control is designed directly using a set of filter systems, with persistent excitation conditions guaranteed in a lower-order manner. For the resulting closed-loop system, we prove that the tracking error converges to zero exponentially. Additionally, a simulation example is provided to demonstrate the effectiveness of the proposed control.

Keywords: Stability Theory
Abstract: This article solves the boundary stabilization problem of variable coefffcient Rayleigh beams by using nonlinear boundary feedback controls. The feedback mechanism incorporates a nonlinear function that adheres to a sector condition, thus encompassing a diverse array of nonlinear feedback scenarios. The well-posedness of the closed-loop system is completed by applying nonlinear semigroup theory. The exponential stability of the energy function of a closed-loop system is demonstrated by using the integral multiplier method.

Keywords: Stability Theory, Actuator and Sensor Placement, Passivity and Dissipativity
Abstract: In this article, we propose a disturbance rejection approach to stabilize 2D nonlinear Kuramoto-Sivashinsky (K-S) equation over a rectangular domain. The control input suffering from the disturbance is actuated through distributed in space characteristic functions. The disturbance is estimated and based on which the disturbance is compensated. Sufficient conditions are explored to ensure the stability of the closed-loop system in terms of the linear matrix inequality (LMI).

Keywords: Stability Theory, Controllability and Observability Analysis, Lyapunov-Based and Backstepping Techniques
Abstract: This paper proposes a novel two-step interval estimation-based state estimation scheme for a class of stochastic parabolic systems. Peak-to-peak analysis is introduced to solve the difficulties generated by the spatiotemporal characteristic and the multidimensional nature. Based on the two-step interval estimation method, the adaptive thresholds of the mathematical expectation of the system state are obtained. Numerical simulation is adopted to show the effectiveness of the proposed method.

Keywords: Controllability and Observability Analysis, Stability Theory
Abstract: We investigate output-to-state stability (OSS) for evolution equations in Banach spaces. We establish the direct OSS Lyapunov theorem and illustrate it with an example. Furthermore, we extend the notion of uniform global asymptotic stability modulo output (UGASMO) to infinite-dimensional systems and show that OSS implies UGASMO, and UGASMO guarantees the vanishing output vanishing state property.

Keywords: Optimal Control, Controllability and Observability Analysis
Abstract: Recently, domain-uniform stabilizability and detectability has been the central assumption to ensure robustness in the sense of exponential decay of spatially localized perturbations in optimally controlled evolution equations. In the present paper we analyze a chain of transport equations with boundary and point controls with regard to this property. Both for Dirichlet and Neumann boundary and coupling conditions, we show a necessary and sufficient criterion on control domains which allow for the domain-uniform stabilization of this equation. We illustrate the results by means of a numerical example.

Keywords: Lyapunov-Based and Backstepping Techniques, Traffic Control and Network Congestion
Abstract: This talk will discuss new methods and perspectives in stabilization. First, we'll look at the stabilization problem for PDEs from an abstract perspective and present an approach called F-equivalence (or sometimes Fredholm backstepping). The principle is simple: instead of directly trying to find a feedback control that makes the system stable, we seek to find a feedback control that transforms the system under consideration into a simpler system, for which stability is already known. We'll come back to the progress made with this method over the last years. Secondly, we'll look at a more concrete problem: the stabilization of hyperbolic equations modeling road traffic. We'll show how abstract mathematical concepts, such as entropic solutions, can have tangible impacts in real-life scenarios, and we'll discuss the application to traffic regulation and the reduction of accordions in traffic jams. Finally, we'll discuss recent advances in AI for mathematics and, in particular, how to train AI to have mathematical intuition to help solve problems.

Keywords: Computational Methods, Experimental Design, Systems Engineering
Abstract: Parameter estimation in differential equation models is a cornerstone in engineering, biology, and beyond. However, limited measurement availability often results in identifiability challenges, making it difficult to recover model parameters accurately. To address this, the formulation of qualitative data can offer a valuable source of supplementary information to constrain the parameter space. In this paper, we formalize parameter estimation for partial differential equations (PDEs) that integrate qualitative constraints from external studies, databases, or prior knowledge. Our approach addresses identifiability issues by effectively narrowing the parameter search space, particularly in scenarios with sparse or incomplete data.

Keywords: Computational Methods, Process Intensification and Process, Energy Generation
Abstract: This work presents the RLS-MAMBA method for lithium-ion battery State of Charge (SOC) estimation. The method integrates Recursive Least Squares (RLS) for dynamic parameter identification with the Mamba neural network for real-time error correction. RLS efficiently updates battery model parameters in real-time, ensuring accurate and adaptive parameter estimation under dynamic operating conditions. The Mamba neural network captures the nonlinear dynamics and long-range dependencies inherent in battery behavior, providing an accurate SOC estimation framework. Experiments with various initial SOC values demonstrate that this method achieves SOC estimation accuracy exceeding 97.56%, with a maximum accuracy of up to 99.98%.

Keywords: Controllability and Observability Analysis, Real-Time Control, Distribution and Storage
Abstract: In this work, we consider the dynamics of repairable systems characterized by three distinct states: one signifying normal operational states, another representing degraded conditions and a third denoting failed conditions. These systems are characterized by their ability to be repaired when failures and/or degradation occur. Typically described by transport equations, these systems exhibit a coupled nature, interlinked through integro-differential equations and integral boundary conditions that dictate the transitions among all the states. In the current work, we mainly address the bilinear controllability of the system via repair actions. Our objective is to enhance the system availability- the probability of being operational when needed over a fixed period of time. We present rigorous analysis and develop control strategies in feedback forms that leverage the bilinear structure of the system model.

Keywords: Distribution and Storage, Flow Control, Model Reduction for Control
Abstract: This paper presents a comprehensive analysis of heat transfer in cooling systems for battery packs in electric vehicles (EVs). A cylindrical coordinate framework is employed to model the heat transfer equation, incorporating diffusion, convection, and heat sources due to electrochemical reactions in the cell. Boundary conditions are derived to account for radial symmetry and cascaded axial heat exchange across modules. The proposed model introduces a coupled nonlinear framework that dynamically interrelates fluid temperature, heat generation, and module temperature, effectively capturing transient and nonlinear thermal interactions.

Keywords: Systems Engineering
Abstract: Driven by the industrial demand for enhanced system safety and control performance, the modeling of distributed parameter systems has drawn significant attention. In this paper, a physics-informed neural network (PINN) fine-tuning scheme, consisting of pre-training and fine-tuning phases, is proposed for modeling distributed parameter systems, while dropout techniques is incorporated to prevent overfitting. During the pre-training phase, the network primarily focuses on training with higher-magnitude loss components, while a variable weighting method adjusts the emphasis according to different training phases. During the fine-tuning phase, the pre-trained model is refined using boundary and initial conditions. Consequently, effective distributed parameter system modeling is achieved for scenarios with a limited number of sensors. The effectiveness of the proposed scheme is demonstrated through a benchmark study on a nonlinear one-dimensional thermal process.

Keywords: Systems Engineering, Controllability and Observability Analysis, Stability Theory
Abstract: This paper addresses the design of a nonlinear observer for an undamped wave equation with distributed disturbance and pointwise position measurement at the boundary. Using the sliding-mode based robust exact differentiator, the velocity at the boundary is estimated. Based on this estimate, a standard Luenberger observer is designed for the nominal, i.e., undisturbed system, and robust convergence is ensured in the presence of bounded distributed disturbances that are constant in time, by adequately combining this observer with an algebraic correction. The proposed design approach achieves exponential convergence of the observer and, for special observer gains, even ensures finite-time convergence. The approach is illustrated using numerical simulations for a representative case study.

Keywords: Semigroup and Operator Theory, Passivity and Dissipativity
Abstract: We characterize the well-posedness of a class of infinite-dimensional port-Hamiltonian systems with boundary control and observation. This class includes in particular the Euler-Bernoulli beam equations and more generally 1D linear infinite-dimensional port-Hamiltonian systems with boundary control and observation as well as coupled systems. It is known, that for the Timoshenko beam models internal well-posedness implies well-posedness of the overall system. By means of an example we show that this is not true for the Euler-Bernoulli beam models. An easy verifiable equivalent condition for well-posedness of the overall system will be presented. We will conclude the paper by applying the obtained results to several Euler-Bernoulli beam models.

Keywords: Passivity and Dissipativity, Computational Methods
Abstract: This paper addresses the port-Hamiltonian modeling of the Rayleigh beam, which bridges the gap between the Euler-Bernoulli and Timoshenko beam theories. This balance makes the Rayleigh model particularly suitable for scenarios where Euler-Bernoulli assumptions are insufficient, but Timoshenko’s complexity is unnecessary, such as in cases of moderate oscillations. The originality of the approach lies in deriving the Rayleigh beam model from the displacement field of the Timoshenko beam and incorporating an algebraic constraint consistent with Rayleigh beam theory. The resulting model is formulated as an infinite-dimensional port-Hamiltonian differential-algebraic equation (PH-DAE). A structure-preserving spatial discretization strategy is developed using the mixed finite element method, ensuring the preservation of the PH-DAE structure in the finite-dimensional setting. Numerical simulations demonstrate the accuracy and effectiveness of the proposed model and discretization approach.

Keywords: Marine Systems and Aerospace Engineering
Abstract: The study of the motion of floating bodies is a question that arises naturally in naval engineering and navigation, but has many applications in different areas of engineering: The study of floating wind turbines and wave energy converters in marine renewable energies, the study of icebergs, etc. It is a question that has been studied since antiquity, but the mathematical formulation of the problem is relatively recent and first studied by Fritz John in two papers (1949, 1950) which, although contain oversimplificating and not entirely correct assumptions, still pose a reference which is studied a lot both from a theoretical and numerical perspective. In these works the fluid is assumed to be irrotational and, hence, the gradient of a velocity potential which becomes the main quantity to describe the motion of the fluid. However, this quantity is not the appropiate object to employ when writing the equations since the domain of the fluid is not smooth, in fact it makes an angle where the fluid is in contact with the body, and the potential becomes singular. Furthermore, the nonlinear effects and the evolution of the boundary of the fluid which is in contact with the solid are missed. The corrected mathematical formulation of the problem can be found in the work by David Lannes, where a different approach is taken to give simple conditions at the so called contact line (the part of the fluid domain which is in contact with the solid) and to derive the evolution law of such line, among other advantages. Another approach to obtain the equations of motion modelling floating bodies is based on the lagrangian formalism of analytical mechanics. Following this method, the aim of this work is to obtain the equations of motion describing the coupled system composed of shallow (ideal) water with a rigid body whose walls are not vertical.

Keywords: Differential Geometric and Algebraic Approaches, Semigroup and Operator Theory
Abstract: In this work, we study the scalar Hodge wave equation on a compact, orientable manifold. We present the necessary differential geometry language to treat Sobolev spaces of differential forms and use these tools to identify a boundary triplet for the problem. We use this boundary triplet to determine a class of boundary conditions for which the problem is well-posed.

Keywords: Passivity and Dissipativity, Marine Systems and Aerospace Engineering, Energy Generation
Abstract: This paper proposes a control-oriented model of a floating wind turbine, incorporating 2D platform motion via a coupled beam-string structure with axial and transversal deformations. The model of the floating turbine includes the rigid body rotations of the floating platform, maintaining the small deformation approximation for the beam. The port-Hamiltonian approach is used for its modularity and to reflect the system's passivity. Simulations using a simplified water-structure interaction modelled by Archimedes’ forces on a rectangular platform are given. Leveraging system modularity, control alternatives are discussed.

Keywords: Smart and Adaptive Structures in Mechatronics, Passivity and Dissipativity, Stability Theory
Abstract: This work introduces a port-Hamiltonian system approach for an optical fiber actuated in two perpendicular directions using a piezoelectric tube, achieving the desired periodic trajectories in both directions. The dynamics of the piezo tube actuator are represented by a finite-dimensional system, while the optical  fiber is represented by an infinite-dimensional one. Additionally, the actuator and optical fiber are interconnected in a power-preserving manner. A control method for output regulation is proposed for the interconnected system, utilizing the internal model principle. The stability analysis of the closed-loop systems is also investigated. The proposed control method is validated through numerical simulations, demonstrating its effectiveness.