arXiv subjects

Guchuan Zhu

Publications and source records attributed to Guchuan Zhu.

At least 19 recordsLinked to original sources

arXiv2026

Input-to-State Stabilization of a Coupled ODE-PDE System with Time-Varying Coefficients via Composite Boundary Control

This paper proposes a novel composite boundary feedback control law that ensures input-to-state stability (ISS) for a coupled ODE-parabolic PDE system with time-varying coefficients in both subsystems. In controller design, we circumvent the need to directly solve coupled time-varying parabolic-hyperbolic kernel equations by employing an analytic pre-defined gain function and a time-varying Volterra kernel function to design the control law explicitly. In stability analysis, to address the simultaneous challenges of Dirichlet boundary disturbances and time-varying coefficients, we employ the square root of a time-varying positive definite matrix and a superlinear function to construct a nonquadratic Lyapunov function for the ODE and a generalized Lyapunov functional for the PDE, respectively, in the target system, thereby establishing the ISS in the $L^2$-norm of the closed-loop system. Numerical simulations are presented to illustrate the effectiveness of the proposed control scheme.

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arXiv2026

Robust synchronization for multi-agent systems governed by PDEs with observable and unobservable disturbances

This paper investigates robust synchronization for multi-agent systems (MASs) governed by parabolic partial differential equations in the presence of both observable and unobservable disturbances. Using only boundary output measurements, a disturbance observer is designed to estimate observable Dirichlet boundary disturbances while ensuring robustness of the observer error system with unobservable disturbances occurring in the domain. Using only the reference signal and local output information, distributed synchronization controllers are then constructed to enable all agents to track the reference trajectory. In particular, exponential tracking is achieved in the absence of unobservable disturbances, while robustness is preserved when additional unobservable disturbances occur during controller implementation. We further analyze the impact of unobservable Dirichlet-Robin boundary disturbances on synchronization performance by proving the boundedness of solutions to the synchronization error system. Moreover, to characterize the influence of all disturbances, input-to-state stability (ISS) is established for the closed-loop system. For the involved systems, the generalized Lyapunov method and the recursion technique are extensively employed in the stability analysis, and the lifting technique and semigroup theory are used to prove the well-posedness. Simulation results validate the proposed control scheme, demonstrating effective disturbance estimation and rejection, robust synchronization, and the ISS properties under various scenarios.

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arXiv2026

Unified Lyapunov Method for ISS of PDEs: A Tutorial on Constructing Generalized Lyapunov Functionals for Parabolic and Hyperbolic Equations

This tutorial provides an overview of the generalized Lyapunov method (GLM) for analyzing input-to-state stability (ISS) of partial differential equations (PDEs). We begin by revisiting the classical Lyapunov method and the standard ISS-Lyapunov theorem, highlighting their limitations when applied to systems with complex boundary disturbances. In contrast, the GLM, based on the concept of generalized Lyapunov functionals (GLFs) that explicitly depend on the external input, offers greater flexibility and efficiency, particularly for PDEs with Dirichlet-type disturbances. The main objective of this tutorial is to demonstrate how to systematically construct GLFs to establish ISS estimates in $L^q$ spaces with any $q\in[2,\infty]$ for different PDEs. Specifically, we consider three representative classes of PDEs: (i) an $N$-dimensional nonlinear parabolic equation with mixed nonlinear boundary disturbances, (ii) a first order nonlinear hyperbolic equation with boundary disturbances, and (iii) a second order linear hyperbolic equation, i.e., a wave equation, with boundary damping and disturbances. For each case, we provide step-by-step constructions of appropriate GLFs and derive explicit ISS estimates, illustrating the general applicability of the GLM. Finally, we discuss open challenges and future directions, including the systematic construction of GLFs for broader classes of PDEs and their applications in controller design.

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arXiv2026

A Spectral-based ISS small-gain theorem for boundary control systems with infinite couplings

We study the input-to-state stability (ISS) of boundary control systems allowing for infinitely many boundary couplings. Using semigroup perturbation theory and the theory of positive linear operators on Banach lattices, we derive a spectral small-gain condition ensuring exponential ISS. We further investigate linear Boltzmann-type equations on an infinite network of intersecting circles, incorporating delays, scattering, and disturbances acting at the junction. For this class of systems, we prove that a spectral small-gain condition on the transmission operator matrix guarantees exponential ISS with respect to disturbances propagating through the network. Moreover, we derive explicit ISS estimates for {certain} classes of dynamical processes. Finally, we demonstrate the practical applicability of our results by considering two important classes of time-delayed transmission conditions.

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arXiv2026

Well-posedness of boundary control systems and application to ISS for coupled heat equations with boundary disturbances and delays

This paper studies the existence of solutions and, in particular, the well-posedness of a class of boundary control systems. Our main result provides explicit and verifiable conditions on the system data that guarantee continuous dependence of solutions on the initial data and $L^p$-inputs. The proof relies on a new boundedness estimate for the input/output maps of linear time-invariant infinite-dimensional systems with unbounded control and observation operators. The developed technique is applied to derive specific conditions for the exponential input-to-state stability of boundary-coupled heat equations with boundary disturbances and time-delays.

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arXiv2026

Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances

This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.

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arXiv2025

Local integral input-to-state stability for non-autonomous infinite-dimensional systems

In this paper, we prove comparison principles for nonlinear differential equations with time-varying coefficients and develop Lyapunov analytical tools for the integral input-to-state stability (iISS) analysis of nonlinear non-autonomous infinite-dimensional systems, which involve nonlinearities satisfying a superlinear growth, {bringing} difficulties to the iISS {analysis.} Specifically, our approach starts by establishing several forms of comparison principles for a wide range of ordinary differential equations having time-varying coefficients and superlinear terms, paving the way to conduct iISS assessment for general nonlinear non-autonomous infinite-dimensional systems within the Lyapunov stability framework. Then, by using the comparison principles, we prove a local {iISS} {(LiISS)} Lyapunov theorem for the nonlinear non-autonomous infinite-dimensional systems in the framework of Banach spaces. {Furthermore,} we provide sufficient conditions of the existence of a local iISS Lyapunonv functional (LiISS-LF) and construct LiISS-LFs for the systems in the framework of Hilbert spaces. Finally, we preset two examples to illustrate the proposed {Lyapunov} method for the LiISS analysis: one is to show how to obtain the LiISS of a nonlinear finite-dimensional system with time-varying coefficients and superlinear terms under linear state feedback control law while another one is to show how to employ the interpolation inequalities to handle superliner terms and establish the LiISS-LF for a class of multi-dimensional parabolic equations with space-time-varying coefficients. To demonstrate the validity of the results, numerical experiments are also conducted to verify the LiISS of these two classes of systems.

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arXiv2024

Finite-time input-to-state stability for infinite-dimensional systems

In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite-dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS-LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS-LF for a class of abstract infinite-dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in-domain disturbances. Since the nonlinear terms of the corresponding abstract system are not Lipschitz continuous, the well-posedness is proved based on the application of compact semigroup theory and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.

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arXiv2024

Input-to-state stabilization of $1$-D parabolic equations with Dirichlet boundary disturbances under boundary fixed-time control

This paper addresses the problem of stabilization of $1$-D parabolic equations with destabilizing terms and Dirichlet boundary disturbances. By using the method of backstepping and the technique of splitting, a boundary feedback controller is designed to ensure the input-to-state stability (ISS) of the closed-loop system with Dirichlet boundary disturbances, while preserving fixed-time stability (FTS) of the corresponding disturbance-free system, for which the fixed time is either determined by the Riemann zeta function or freely prescribed. To overcome the difficulty brought by Dirichlet boundary disturbances, the ISS and FTS properties of the involved systems are assessed by applying the generalized Lyapunov method. Numerical simulations are conducted to illustrate the effectiveness of the proposed scheme of control design.

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arXiv2024

Input-to-State Stabilization of 1-D Parabolic PDEs under Output Feedback Control

This paper addresses the problem of input-to-state stabilization for a class of parabolic equations with time-varying coefficients, as well as Dirichlet and Robin boundary disturbances. By using time-invariant kernel functions, which can reduce the complexity in control design and implementation, an observer-based output feedback controller is designed via backstepping. By using the generalized Lyapunov method, which can be used to handle Dirichlet boundary terms, the input-to-state stability of the closed-loop system under output feedback control, as well as the state estimation error system, is established in the spatial $L^\infty$-norm. Numerical simulations are conducted to confirm the theoretical results and to illustrate the effectiveness of the proposed control scheme.

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arXiv2023

Power Tracking Control of Heterogeneous Populations of TCLs with Partially Measured States

This paper presents a new aggregate power tracking control scheme for populations of thermostatically controlled loads (TCLs). The control design is performed in the framework of partial differential equations (PDEs) based on a late-lumping procedure without truncating the infinite-dimensional model describing the dynamics of the TCL population. An input-output linearization control scheme, which is independent of system parameters and uses only partial state measurement, is derived, and a sliding model-like control is applied to achieve finite-time input-to-state stability for tracking error dynamics. Such a control strategy can ensure robust performance in the presence of modeling uncertainties, while considerably reducing the communication burden in large scale distributed systems similar to that considered in the present work. A rigorous analysis of the closed-loop stability of the underlying PDE system was conducted, which guaranteed the validity of the developed control scheme. Simulation studies were performed while considering two TCL populations with a significant difference in their size, and the results show that the developed control scheme performs well in both cases, thereby confirming the effectiveness of the proposed solution.

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arXiv2023

Power Tracking Control of Heterogeneous TCL Populations with Modeling Uncertainties and Communication Restrictions

This paper presents a new aggregate power tracking control scheme for populations of thermostatically controlled loads (TCLs). The control design is carried out in the framework of partial differential equations (PDEs) based on a late-lumping procedure without truncating the infinite-dimensional model describing the dynamics of the TCL population. An input-output linearization control scheme, which is independent of the system parameters and uses only partial state measurement, is derived, and a sliding model control is applied, which allows achieving a finite-time input-to-state stability for the tracking error dynamics. Such a control strategy can ensure a robust performance in the presence of modeling uncertainties while considerably reducing the communication burden in large scale distributed systems as the one considered in the present work. To guarantee the validity of the developed control scheme, a rigourous analysis on the solutions to the underlying PDE is conducted. Two implementations of the proposed control strategy, based on discrete-time approximation and fuzzy logic control, respectively, are validated through simulation studies.

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arXiv2023

Proof of Proposition 3.1 in the paper titled "Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions''

We provide a detailed proof of Proposition 3.1 in the paper titled ``Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions''. In the paper titled ``Backstepping control of a class of space-time-varying linear parabolic PDEs via time invariant kernel functions'', we addressed the problem of exponential stabilization and continuous dependence of solutions on initial data in different norms for a class of $1$-D linear parabolic PDEs with space-time-varying coefficients under backstepping boundary control. In order to stabilize the system without involving a Gevrey-like condition or the event-triggered scheme, a boundary feedback controller was designed via a time invariant kernel function. By using the approximative Lyapunov method, the exponential stability of the closed-loop system was established in the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, whenever $p\in [1, +\infty]$. It was also shown that the solution to the considered system depends continuously on the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, of the initial data.

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arXiv2022

Exponential stabilization and continuous dependence of solutions on initial data in different norms for space-time-varying linear parabolic PDEs

For an arbitrary parameter $p\in [1,+\infty]$, we consider the problem of exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm, respectively, for a class of anti-stable linear parabolic PDEs with space-time-varying coefficients in the absence of a Gevrey-like condition, which is often imposed on time-varying coefficients of PDEs and used to guarantee the existence of smooth (w.r.t. the time variable) kernel functions in the literature. Then, based on the obtained exponential stabilities, we show that the solution of the considered system depends continuously on the $L^{p}$-norm, and $W^{1,p}$-norm, respectively, of the initial data. In order to obtain time-independent (and thus sufficiently smooth) kernel functions without a Gevrey-like condition and deal with singularities arising in the case of $p\in[1,2)$, we apply a combinatorial method, i.e., the combination of backstepping and approximation of Lyapunov functionals (ALFs), to stabilize the considered system and establish the continuous dependence of solutions on initial data in different norms.

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arXiv2021

Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs

In this paper, we introduce the notion of relative $\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous dependence of (weak) solutions on external disturbances for nonlinear parabolic PDE systems. Based on the RKES, we prove the input-to-state stability (ISS) in the spatial sup-norm for a class of nonlinear parabolic PDEs with either Dirichlet or Robin boundary disturbances. Two examples, concerned respectively with a super-linear parabolic PDE with Robin boundary condition and a $1$-D parabolic PDE with a destabilizing term, are provided to illustrate the obtained ISS results. Besides, as an application of the notion of RKES, we conduct stability analysis for a class of parabolic PDEs in cascade coupled over the domain or on the boundary of the domain, in the spatial and time sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi iteration is extensively used in the proof of the results presented in this paper.

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arXiv2020

A PDE--based Power Tracking Control of Heterogeneous TCL Populations

This chapter presents the development and the analysis of a scheme for aggregate power tracking control of heterogeneous populations of thermostatically controlled loads (TCLs) based on partial differential equations (PDEs) control theory and techniques. By employing a thermostat--based deadband control with forced switching in the operation of individual TCLs, the aggregated dynamics of TCL populations are governed by a pair of Fokker--Planck equations coupled via the actions located both on the boundaries and in the domain. The technique of input-output feedback linearization is used for the design of aggregate power tracking control, which results in a nonlinear system in a closed loop. As the considered setting is a problem with time-varying boundaries, well-posedness assessment and stability analysis are carried out to confirm the validity of the developed control scheme. A simulation study is contacted to evaluate the effectiveness and the performance of the proposed approach.

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arXiv2020

Approximations of Lyapunov functions for ISS analysis of a class of nonlinear parabolic PDEs

This paper addresses the input-to-state stability (ISS) and integral input-to-state stability (iISS) for a class of nonlinear higher dimensional parabolic partial differential equations (PDEs) with different types of boundary disturbances (Robin or Neumann or Dirichlet) from different spaces by means of approximations of Lyapunov functions. Specifically, by constructing approximations of (coercive and non-coercive) ISS Lyapunov functions we establish: (i) the ISS and iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary disturbances from $L^q_{loc}(\mathbb{R}_+;L^1(\partialΩ))$-space for any $q\in [1,+\infty]$; (ii) the iISS in $L^1$-norm (and weighted $L^1$-norm) for PDEs with boundary disturbances from $L^Φ_{loc}(\mathbb{R}_+;L^1(\partialΩ))$-space for certain Young function $Φ$; and (iii) the ISS and iISS in $L^Φ$-norm (and weighted $K_Φ$-class) for PDEs with boundary disturbances from $L^q_{loc}(\mathbb{R}_+;K_Φ(\partialΩ))$-class for any $q\in [1,+\infty]$ and certain Young function $Φ$. The ISS properties stated in (ii) and (iii) are assessed in the framework of Orlicz space or Orlicz class.

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arXiv2020

A Note on the Maximum Principle-based Approach for ISS Analysis of Higher Dimensional Parabolic PDEs with Variable Coefficients

This paper presents a maximum principle-based approach in the establishment of input-to-state stability (ISS) for a class of nonlinear parabolic partial differential equations (PDEs) over higher dimensional domains with variable coefficients and different types of nonlinear boundary conditions. Technical development on ISS analysis of the considered systems is detailed, and an example of establishing ISS estimates for a nonlinear parabolic equation with, respectively, a nonlinear Robin boundary condition and a nonlinear Dirichlet boundary condition is provided to illustrate the application of the developed method.

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