Keywords: Semigroup and Operator Theory, Passivity and Dissipativity
Abstract: We review some known and some less known theorems for the local (in time) representation of distributed parameter systems in the Hilbert space context. We use the Crandall-Pazy theory of monotone operators and nonlinear semigroups of contractions.

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.

Keywords: Stability Theory
Abstract: This work is concerned with boundary stabilization of multi-dimensional discrete velocity kinetic models. By exploiting a certain stability structure of the models and adapting an appropriate functional, we derive feasible control laws so that the corresponding solutions decay exponentially in time. The result is illustrated with an application to the two-dimensional model in a square container. This is a joint work with Haitian Yang.

Keywords: Stability Theory
Abstract: In recent years, the stabilization of the network of Saint-Venant equations arises due to the application requirements. In this talk, we focus on a cascade as well as star-shape like network of open channels with interface conditions modeled by general Saint-Venant equations with arbitrary cross section, slope and friction. The control input and measured output are both on the collocated boundary. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the non- uniform steady-states of different channels is constructed. We show that by a suitable choice of the boundary feedback controls, the local exponential stability of the nonlinear Saint-Venant equations for the H2 norm are guaranteed.

Keywords: Stability Theory
Abstract: In this talk, we are concerned with finite-time boundary stabilization of 1D linear coupled hyperbolic systems by using backstepping approach. These systems may involve hyperbolic balance laws, hyperbolic partial integro-differential equations and related non-autonomous systems, which have a wide range of physical engineering applications. We will show how to use Volterra and in- vertible Fredholm transformations to reduce the internal coupling terms of these initial systems as much as possible. Such a reduction makes it trivial to see that the corre- sponding target systems decay to zero in optimal finite time. In particular, the boundary feedback controls can be applied to one side or both sides of the systems.

Keywords: Stability Theory
Abstract: This talk is about the rapid stabilization of general linear systems, when the differential operator A has a Riesz basis of eigenvectors. We find simple sufficient conditions for the rapid stabilization and the construction of a relatively explicit feedback operator. We use an F- equivalence approach relying on Fredholm transformation to show a stronger result: under these sufficient conditions the system is equivalent to a simple exponentially stable system, with arbitrarily large decay rate. In particular, our conditions improve the existing conditions of rapid stabilization for non-parabolic operators such as skew- adjoint systems.

Keywords: Lyapunov-Based and Backstepping Techniques
Abstract: Time delays arising from the transport of material, information, or energy frequently degrade control performance and may destabilize dynamical systems. The PDE backstepping method has emerged as an effective approach for the design of controllers and observers in distributed parameter systems influenced by delays.

Delays can be classified based on the affected system components (input, measurement, or state delays), their characteristics (unknown, stochastic, temporally varying, or spatially varying), and intrinsic system properties such as dimensionality and PDE type (e.g., parabolic or hyperbolic). To incorporate delays within the PDE backstepping control framework, transport PDEs are often introduced to represent delayed variables, cascading with the original system dynamics. This yields distinct coupled systems: parabolic–hyperbolic cascades for parabolic plants and augmented hyperbolic systems for first-order hyperbolic plants.

Building upon our work on distributed input delays, multi-delay scenarios, spatially varying delays (continuum multi-delay), and two-dimensional parabolic systems subject to delays, this talk focuses on common features in the backstepping design of predictive controllers. In particular, we analyze the connections and distinctions in backstepping transformations, characteristics of kernel equations, and corresponding controller structures for various PDE types and delay forms.

Keywords: Stability Theory, Semigroup and Operator Theory, Controllability and Observability Analysis
Abstract: This work studies the boundary feedback stabilization for an elastic thin plate, where one part of its boundary is clamped and the remaining free part is attached to a rigid body. The boundary feedback controls are applied to the free boundary. Given that the boundary of the domain may have corners, we introduce a proper abstract framework to analyze the well-posedness and estimate the resolvent. When the geometric control condition (GCC) holds, we prove a polynomial energy decay rate of t^{-{1over2}}. For cases where GCC does not hold, we analyze the system on a rectangular domain with control applied to only one edge and obtain a slower decay rate t^{-{1over3}}.

Keywords: Flow Control, Computational Methods, Flexible Micro-Structures
Abstract: The article examines the pressure-controlled dynamics of the flowrate in an elastic microchannel. As shown by numerical simulations, the elasticity of the channel gives rise to a complex and nonlinear transient response. The employed numerical scheme is based on the method of characteristics. It is also used to solve the inverse problem of determining the inlet pressure required to achieve a smooth flowrate at the microchannel outlet.

Keywords: Stability Theory, Semigroup and Operator Theory
Abstract: This study investigates the optimal decay rate of a degenerate coupled heat-wave system within two interconnected bounded intervals. In Han et al. (2023), the authors delved into the long-time behavior of such a coupled system and established that the solutions to this system converge to zero with an explicit decay rate, which is dependent on the degree of degeneration of the diffusion coefficient in the heat equation near the interface. However, the optimality of this derived decay rate remained unresolved. The present work aims to affirm the optimality of this explicit decay rate through a meticulous asymptotic spectral analysis of the system operator by the properties of Bessel functions, complemented by an examination of the frequency characteristics of the semigroup.

Keywords: Computational Methods, Smart and Adaptive Structures in Mechatronics, Semigroup and Operator Theory
Abstract: An inversion-based approach for the Euler-Bernoulli beam modeled in terms of a boundary controlled port-Hamiltonian system is presented. The goal is to achieve an open-loop finite-time transition between steady states. Exchanging the systems input by a new (fictitious) boundary condition in terms of a so-called basic output located at the boundary or inside the spatial domain, the port-Hamiltonian system is reformulated as a boundary value problem in the spatial domain. Input solution samples are numerically calculated with an inverse Laplace transformation or Fast Fourier Transformation algorithm by assigning a suitable desired trajectory for the basic output. The presented solution approach is evaluated by numerical calculations and simulations.

Keywords: Lyapunov-Based and Backstepping Techniques, Stability Theory, Real-Time Control
Abstract: This paper develops a bilateral delay-compensation control strategy for an unstable 2-D reaction-diffusion system with unequal input delays. The main novelty compared to our previous work lies in the design of asymmetric backstepping transformations tailored to the unequal delays, which results in a more complex kernel equation associated with the larger delay and poses significant challenges in proving its well-posedness. To address this, we employ a combination of operator semigroup theory and Lyapunov function analysis. Based on this result, we prove the boundedness and invertibility of the backstepping transformations and establish the exponential stability of the closed-loop system under the proposed compensator.

Keywords: Model Reduction for Control, Controllability and Observability Analysis, Computational Methods
Abstract: We study the spectral pollution detected in the spectrum of Gramians associated with the controlled parabolic systems. We improve the upper spectral bounds that guarantee the occurrence of the phenomenon for any kind of finite dimensional approximation of the original problem. Furthermore, we provide a criterion to determine the pollution free part of the spectrum.

Keywords: Semigroup and Operator Theory, Passivity and Dissipativity
Abstract: We provide a framework for the well-posedness of a feedback system obtained by closing a feedback loop around an incrementally impedance passive, possibly nonlinear system. The static feedback operator is assumed to be maximal monotone. Our approach uses the theory of maximal monotone operators and Lax-Phillips type nonlinear semigroups.

Keywords: Semigroup and Operator Theory, Passivity and Dissipativity, Stability Theory
Abstract: We study the output regulation of a passive system using collocated feedback for strong stabilization, and the classical resonant internal model based controller. We prove that this controller solves the output regulator problem if the concept of ``the error converges to zero'' is defined in a suitable way: low-pass filtering of the error with any low-pass filter gives a continuous signal that tends to zero.

Keywords: Stability Theory
Abstract: In the present work, we focus on the study of stabilization of 1-D nonlinear KdVB equation via modal decomposition approach, under one or several distributed in space point actuators in the presence of variable input delay.