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: Computational Methods, Lyapunov-Based and Backstepping Techniques
Abstract: Control of distributed parameter systems affected by delays is a challenging task, particularly when the delays depend on spatial variables. The idea of integrating analytical control theory with learning-based control within a unified control scheme is becoming increasingly promising and advantageous. In this paper, we design a integrated control strategy combining PDE backstepping and deep reinforcement learning (RL) for an unstable first-order hyperbolic PDE with spatially-varying delays. This method eliminates extra constraint on the delay function required for the backstepping design. We embed a DeepONet, trained to learn the backstepping controller, into a soft actor-critic (SAC) framework as a feature extractor for both the actor and critic networks. Simulation results demonstrate that the proposed algorithm outperforms standard SAC in reducing steady-state error and surpasses the backstepping controller in mitigating overshoot.

Keywords: Process Intensification and Process, Systems Engineering, Passivity and Dissipativity
Abstract: Significant contributions have been made in continuous-time thermodynamic modeling, analysis, control, and estimation of distributed parameter systems. Compared with continuous-time works, research on discrete-time approaches is relatively limited. This contribution studies discrete-time modeling and analysis of a class of linear thermodynamic distributed parameter systems described by coupled hyperbolic partial differential equations (PDE) that can describe various applications in practice. Based on the Cayley-Tustin time discretization approach, the continuous-time infinite-dimensional model is converted to a discrete-time infinite dimensional model, where no spatial discretization or model order reduction is performed. We show that under this transformation, an approximation of the total internal entropy production is preserved, and the difference between the supply rates of the continuous- and discrete-time systems converges to zero as the time discretization interval tends to zero. Finally, the proposed analysis is illustrated through a counter-current heat exchanger with two state variables.

Keywords: Distributed Cooperative Systems, Systems Engineering, Computational Methods
Abstract: Distributed parameter systems (DPSs) are prevalent in various industrial processes. However, spatiotemporal coupling, nonlinearity, and time-varying dynamics lead to complex modeling of DPSs. Traditional approaches often fail to address these challenges effectively. This paper proposes a collaborative three-dimensional (3D) fuzzy modeling framework that integrates automatic clustering based on an adaptive genetic algorithm and support vector regression (SVR) for complex DPS modeling. The proposed method constructs a collaborative 3D fuzzy model by utilizing automatic clustering to extract fuzzy rules and employing SVR to learn spatial basis functions while incorporating spatiotemporal separation and synthesis. Experimental validation of the rapid thermal chemical vapor deposition heating process demonstrates the effectiveness and superiority of the proposed approach.

Keywords: Lyapunov-Based and Backstepping Techniques
Abstract: In nonlinear ODE-PDE coupled systems, where propagation speed depends on the integral of ODE states over a specified past period, we develop a trilayer predictor-feedback control strategy to fully compensate for input delays associated with the double integral of the ODE states. Assuming bounded transport velocity, we establish global asymptotic stability of the closed-loop system in the infinity norm through a predictor-based backstepping transformation. We model a buffer driven production system and perform numerical experiments to demonstrate the effectiveness of our control approach using a previously introduced piecewise exponential bang bang control law as the nominal (delay-free) feedback design.

Keywords: Optimal Control, Stability Theory, Actuator and Sensor Placement
Abstract: We studies a data-driven optimal boundary control method for diffusion-reaction processes coupled with actuator dynamics. Considering the actuator characteristics as ordinary differential equations, we first rewrite the coupled distributed parameter system with boundary control as one with distributed control inputs. Subsequently, applying Bellman optimality principle, a novel policy iterative optimal control algorithm is designed, which collects state datasets along the system trajectory and then iteratively computes the optimal control policy offline. There is no need to run the system online and the collected dataset can be reused, greatly saving computational resources.

Keywords: Lyapunov-Based and Backstepping Techniques, Adaptive Optics
Abstract: Existing results on adaptive boundary control for hyperbolic PDE systems based on regular gradient descent or least squares laws have two general limitations, namely, there is no theoretical guarantee for the convergence of parameter estimation error and only asymptotic system stability is achieved. This paper proposes a novel adaptive controller for stabilizing 2times 2 hyperbolic PDE systems with uncertain distal reflection. As only the boundary state is measurable, an adaptive observer is designed to estimate the system states and the boundary uncertainty, simultaneously. For the boundary uncertainty, we develop a new parameter estimation law, called the concurrent learning parameter estimation law, that simultaneously leverages the real-time and historical system states to update the estimated parameter. It is shown that the estimation errors of the system states and the uncertain parameter are exponentially convergent. Then, an adaptive output-feedback backstepping controller is developed based on the observed states and the estimated parameter. Thanks to the accurate parameter estimation, the controller guarantees exponential system stability rather than asymptotic stability. Finally, the proposed control scheme is demonstrated in a numerical comparison simulation.