Keywords: Traffic Control and Network Congestion, Stability Theory
Abstract: Modern urban mobility systems exhibit inherently complex and multi-scale dynamics where macroscopic traffic flow interacts with microscopic vehicle behaviors, shaping system-wide performance in congestion, efficiency, and safety. Emerging technologies—autonomous driving, wireless communication, and AI —are transforming urban mobility, from smart infrastructure management to connected autonomous vehicles (CAVs) design. This talk explores how control theory can bridge the macro-micro loop to optimize urban mobility systems. The talk will begin with infrastructure-based traffic control, where macroscopic sensing and actuation (e.g., variable speed limits, ramp metering) are designed using Partial Differential Equation (PDE)-based control frameworks to mitigate stop-and-go traffic congestion. The we will discuss how CAVs enable fine-grained sensing and control at the microscopic level. Compared with human-driven vehicles (HVs), enhanced perception and control capabilities of CAVs unlock new opportunities for traffic stabilization and safety-guarantee. We present a safety-critical traffic control design using control barrier functions, ensuring collision-free guarantees in mixed-autonomy traffic. The talk also highlights how physics-informed learning and neural operator learning can further enhance PDE control designs, merging theoretical rigor with data-driven adaptability for next-generation mobility systems.

Keywords: Lyapunov-Based and Backstepping Techniques, Controllability and Observability Analysis, Stability Theory
Abstract: In this paper, we design a finite-time convergent observer for linear time-varying ODE-hyperbolic balance law cascade systems. This work extends the study of output regulation for linear time-varying hyperbolic systems. In Bai et al. (2025), a state feedback regulator was developed to achieve finite-time output regulation under the assumption that all disturbances and system states are known. However, this assumption is unrealistic in practical applications. Therefore, we design a finite-time observer under the condition that only the to-be-tracked reference signal and boundary values of the system state are the available measurement. The key idea is to use two ODE observers to achieve finite-time observation of the ODE subsystem and to exploit the finite-time stability structure of the hyperbolic subsystem to estimate its state. In this work, we utilize results on the observability and estimability of linear time-varying ODE systems to establish the existence of the proposed observer.

Keywords: Lyapunov-Based and Backstepping Techniques, Semigroup and Operator Theory
Abstract: We develop a boundary observer design for a class of continua of linear hyperbolic PDE systems, which are viewed as the continuum version of n+m general heterodirectional hyperbolic systems as ntoinfty. The design relies on the introduction of a novel, continuum PDE backstepping transformation, which enables the construction of a Lyapunov functional for the estimation error system. We then introduce the respective non-collocated, output-feedback design employing the stabilizing continuum control kernels from cite{HumBek24arxivc}. (The observer-based output-feedback stabilization problem for the class of infty+1 hyperbolic systems can be solved as special case of the design procedure presented here.) Stability under the observer-based output-feedback law is established by using the Lyapunov functional construction for the estimation error system and proving well-posedness of the complete closed-loop system, which allows utilization of the separation principle. We illustrate the output-feedback design in simulation, via a numerical example for which the control and observer kernels can be computed in closed form.

Keywords: Lyapunov-Based and Backstepping Techniques, Stability Theory
Abstract: To stabilize a 2×2 first-order hyperbolic partial differential equation system with input delay, a backstepping-based feedback controller requires kernel functions as control gains. These gains are obtained by mapping system parameters into spatial-domain functions via nonlinear operators. Due to the delay, multiple layers of mapping exist among the kernel functions, increasing the complexity of computing the control gains. Neural operators efficiently approximate these kernel functions without explicitly solving the associated equations. In this paper, we develop DeepONet-based neural operators and show that the closed-loop system remains exponentially stable with adjustable decay rates when using approximate gains. Numerical simulations illustrate the theoretical results and demonstrate the robustness of the closed-loop system with neural operators. Additionally, the simulations quantify a computational speedup of at least two orders of magnitude compared to traditional PDE solvers.

Keywords: Lyapunov-Based and Backstepping Techniques, Stability Theory
Abstract: In this paper, we propose a feedback boundary control law that ensures the exponential stability of a class of hyperbolic systems characterized by a non-diagonalizable principal part and nonuniform coefficients, as encountered in sedimentation processes and elasticity theory. We establish sufficient conditions for the exponential stability of the closed- loop system. A numerical example is presented to illustrate the effectiveness of the proposed Lyapunov-based stability analysis.

Keywords: Stability Theory
Abstract: We design a finite-dimensional state-feedback controller for the semilinear wave equation with in-domain damping under Neumann actuation. We employ the dynamic extension to shift control from the boundary to the main equation and design a controller that stabilizes a finite number of dominating modes. To address potential instability caused by the remaining modes (spillover), we develop and compare two methods: the L^2 residue separation method, resulting in an algebraic Riccati equation, and a method based on Young's inequality, leading to linear matrix inequalities (LMIs). Numerical results show that the L^2 residue separation method provides less conservative conditions, allowing larger Lipschitz constants compared to Young's inequality approach.

Keywords: Lyapunov-Based and Backstepping Techniques, Stability Theory
Abstract: We consider a class of mixed-coupling hyperbolic-parabolic partial differential equation (PDE) systems, motivated by the melt spinning process in chemical fiber industry. This system involves a new type of coupling, where the derivative term of the hyperbolic state is coupled into the parabolic equation. This is a distinction from existing studies, which primarily consider source term or integral term coupling. In the paper, we propose a two-step backstepping transformation and design two boundary controllers: one for the parabolic PDE and the other for the hyperbolic PDE. We prove the well-posedness of the coupled kernel equations using an infinite induction energy series approach. Additionally, we demonstrate the invertibility of the transformation and establish the exponential stability of the closed-loop system in the L2 norm. Finally, the theoretical results are validated through numerical simulations.

Keywords: Quantum Systems
Abstract: Quantum dynamics provides the arguably most fundamental example of hybrid dynamics: As long as no measurement takes place, the system state is governed by the Schrödinger-Liouville differential equation, which is however interrupted and replaced by projective dynamics at times when measurements take place. We show how this alternatingly continuous and projective evolution can be cast in form of one single differential equation for a refined state space manifold and thus be made amenable to standard port-theoretic analysis and control techniques.

Keywords: Semigroup and Operator Theory, Stability Theory
Abstract: This paper deals with the problem of designing an estimator-based H_infty boundary controller for semilinear parabolic It^ o-type stochastic partial differential systems suffered from the intrinsic random fluctuation, external random disturbance, and stochastic measurement noise. A state estimator is first constructed for the stochastic system by using the collocated boundary observation. On the basis of Lyapunov technique, an estimator-based H_infty boundary controller is then designed guaranteeing closed-loop mean-square exponential stability of the augmented system with an H_infty control performance. Well-posedness and stability analysis of a mild solution for the disturbance-free closed-loop augmented system are deduced via C_0-semigroup approach. Finally, a numerical example is given to confirm the performance of the developed approach.

Keywords: Stability Theory
Abstract: This paper introduces an intermittent sampled-data control issue for coupled systems represented by ordinary differential equations (ODEs) and partial differential equations (PDEs). First, the exponential stability conditions in the form of linear matrix inequalities (LMIs) for coupled PDE-ODE systems under intermittent sampled-data controller are given by using switched time-dependent Lyapunov functional. Then, intermittent sampled-data control gain is obtained by solving LMIs. Finally, the proposed design strategy is successfully applied to the control problem of a hypersonic rocket car (HRC).

Keywords: Stability Theory, Distributed Cooperative Systems, Lyapunov-Based and Backstepping Techniques
Abstract: A coordination control algorithm is presented to solve the angle coordination and vibration control problems of the master-slave flexible manipulators with bending-twisting deformation. Considering the effects of bending deformation and torsional deformation, the partial differential equation model of the flexible manipulators are established based on Hamilton principle. Through the coordination control strategy proposed in this paper, the master's joint angle can track the ideal target angle and the joint angle of slave manipulator can track the master’s angle. In addition to angle tracking, the proposed controller can also reduce the vibration-induced elastic deformation of the manipulators. The closed-loop system can be proved to be exponentially stable via the Lyapunov direct method. Finally, simulation examples verify the satisfactory performance of the coordination control algorithm.

Keywords: Systems Engineering, Semigroup and Operator Theory, Passivity and Dissipativity
Abstract: This paper addresses the stabilization problem of conservative ODE-PDE systems with input saturation. We first introduce the main theoretical framework for abstract conservative systems and then apply this approach to a specific gantry crane system. Specifically, we propose a nested saturation control method that utilizes partial state measurements, leveraging a weak Lyapunov functional and LaSalle's Invariance Principle to achieve asymptotic stabilization of the closed-loop system.