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: 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.

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.