arXiv subjects

Hugo Lhachemi

Publications and source records attributed to Hugo Lhachemi.

At least 19 recordsLinked to original sources

arXiv2026

Controllability of wave-heat and heat-wave cascades

We study boundary controllability of one-dimensional coupled hyperbolic-parabolic cascades, focusing on the fine structure of reachable sets. The main model is a wave-heat cascade in which a boundary control acts on the wave equation and drives the heat equation through an internal coupling. We provide a sharp minimal time for the hyperbolic part (T > 2L) and a complete spectral characterization of exact controllability in weighted Hilbert spaces, whose definition depends explicitly on the coupling profile through a sequence of modal coefficients. In particular, internal couplings may generate nonstandard highly irregular controllability spaces and yield a generic (full measure) but non-robust controllability property. The analysis relies on Riesz basis decompositions and on an Ingham-M{\"u}ntz inequality. We also prove that the exact controllability space is not invariant along Hilbert Uniqueness Method trajectories: even if both endpoints belong to the controllability space, the associated minimal-energy trajectory may leave it at intermediate times. Finally, we compare with the reversed (heat-wave) cascade and discuss how reversing the direction of the coupling transfers the loss of regularity between the parabolic and hyperbolic components.

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arXiv2026

Stabilization of a Wave-Heat Cascade System

We consider the output-feedback stabilization of a one-dimensional cascade coupling a reaction-diffusion equation and a wave equation through an internal term, with Neumann boundary control acting at the wave endpoint. Two measurements are available: the wave velocity at the controlled boundary and a temperature-type observation of the reaction-diffusion component, either distributed or pointwise. Under explicit, necessary and sufficient conditions on the coupling and observation profiles, we show that the generator of the open-loop system is a Riesz-spectral operator. Exploiting this structure, we design a finite-dimensional dynamic output-feedback law, based on a finite number of parabolic modes, which achieves arbitrary exponential decay in both the natural energy space and a stronger parabolic norm. The construction relies on a spectral reduction and a Lyapunov argument in Riesz bases. We also extend the design to pointwise temperature or heat-flux measurements.

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arXiv2025

Controllability and Stabilization of a Wave-Heat Cascade System

Considering a wave-reaction-diffusion PDE cascade system with wave Neumann control, we first establish controllability properties in a suitable Hilbert space depending on the coupling cascade term. This is done by deriving an observability inequality for the dual problem by resorting to an Ingham-M{\"u}ntz inequality. Second, we design an explicit output feedback control strategy for the actual stabilization of the PDE cascade. The key property is that the underlying operator is a Riesz-spectral operator.

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arXiv2025

Boundary control of heat-heat cascades

This paper addresses the problem of feedback stabilization of a cascade of two heat equations that are coupled in the boundary conditions, the input being a boundary control for the first component of the cascade. Two distinct control input settings are studied: one being collocated with the coupling condition of the two heat equations, and the other being noncollocated. These two different configurations induce different controllability properties. The key idea developed in this paper is to carry out spectral reductions, not for each of the two components of the cascade separately, but instead, directly for the PDE cascade viewed as one single system. A detailed study of the eigenelements of the PDE cascade yields a complete characterization of the spectral mode controllability and allows us to derive an explicit state-feedback control strategy for the exponential stabilization of the plant. This approach is extended to a systematic output-feedback control strategy either with a distributed output operator or with a pointwise measurement done on the second heat equation of the PDE cascade. In both state-feedback and output-feedback scenarios, stabilization results are established in L^2 and H^1 norms. Finally, we show how the results developed in this paper for the two studied heat-heat cascades extend to their dual problems.

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arXiv2025

Boundary output feedback stabilization of a cascade of N heat equations

This paper solves the problem of output feedback stabilization for a cascade of N heat equations that are coupled at the boundary, the input being a scalar boundary control applied to the first heat equation of the cascade, and the scalar output being either a distributed or a pointwise in-domain measurement done on the last equation of the cascade. Two different configurations are studied in details. The first one consists of a cascade of N heat equations with totally disconnected spectra. The second one consists of a cascade of N identical heat equations, inducing eigenvalues of multiplicity N . In both cases, the problem is solved thanks to a spectral analysis and a study of the modal controllability and observability properties. The key point is that the generalized eigenvectors form a Riesz basis of the state space. The stabilization property is established in L^2 and H^1 norms.

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arXiv2023

Design of low-dimensional controllers for high-dimensional systems

This article presents proposals for the design of reduced-order controllers for high-dimensional dynamical systems. The objective is to develop efficient control strategies that ensure stability and robustness with reduced computational complexity. By leveraging the concept of partial pole placement, which involves placing a subset of the closed-loop system's poles, this study aims to strike a balance between reduced-order modeling and control effectiveness. The proposed approach not only addresses the challenges posed by high-dimensional systems but also provides a systematic framework for controller design. When an infinite-dimensional operator is Riesz spectral, our theoretical analysis highlights the potential of partial pole placement in advancing control design. Model uncertainties, introduced by an error on the spectral decomposition, can also be allowed. This is illustrated in particular in the case of systems modeled by coupled ordinary-partial differential equations (ODE-PDE).

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arXiv2023

Instability conditions for reaction-diffusion-ODE systems

This paper analyzes the stability of a reactiondiffusion equation coupled with a finite-dimensional controller through Dirichlet boundary input and Neumann boundary output. Going against the flow, we intend to propose numerical certificates of instability for such interconnections. From one side, using spectral methods, an analytical condition based on root locus analysis can determine the instability regions in the parameters space and can sometimes be tested. On the other side, using Lyapunov direct and converse approaches, two sufficient conditions of instability are established in terms of linear matrix inequalities. The novelties lie both in the type of system studied and in the methods used. The numerical results demonstrate the performance of the different criteria set up in this paper and allow us to conjecture that these conditions seem to be necessary and sufficient.

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arXiv2023

Boundary output feedback stabilisation for 2-D and 3-D parabolic equations

The present paper addresses the topic of boundary output feedback stabilization of parabolic-type equations, governed by linear differential operators which can be diagonalized by the introduction of adequate weighting functions (by means of the Sturm-Liouville method), and which evolve in bounded spatial domains that are subsets of $\mathbb{R}^d,\ d=1,2,3$. Combining ideas inspired by \cite{lhachemi2022finite} for the boundary output feedback control of 1-D parabolic PDEs and \cite{munteanu2019boundary} for the state feedback control of multi-D parabolic PDEs, we report in this paper an output feedback boundary stabilizing control with internal Dirichlet measurements designed by means of a finite-dimensional observer. The reported control design procedure is shown to be systematic for 2-D and 3-D parabolic equations.

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arXiv2022

Local output feedback stabilization of a Reaction-Diffusion equation with saturated actuation

This paper is concerned with the output feedback stabilization of a reaction-diffusion equation by means of bounded control inputs in the presence of saturations. Using a finite-dimensional controller composed of an observer coupled with a finite-dimensional state-feedback, we derive a set of conditions ensuring the stability of the closed-loop plant while estimating the associated domain of attraction in the presence of saturations. This set of conditions is shown to be always feasible for an order of the observer selected large enough. The stability analysis relies on Lyapunov functionals along with a generalized sector condition classically used to study the stability of linear finite-dimensional plants in the presence of saturations.

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arXiv2021

Local Output Feedback Stabilization of a Nonlinear Kuramoto-Sivashinsky equation

This paper is concerned with the local output feedback stabilization of a nonlinear Kuramoto-Sivashinsky equation. The control is located at the boundary of the domain while the measurement is selected as a Neumann trace. This choice of system output requires the study of the system trajectories in $H^2$-norm. Moreover, the choice of the actuation/sensing scheme is discussed and adapted in function of the parameters of the plant in order to avoid the possible loss of controllability/observability property of certain eigenvalues of the underlying operator. This leads in certain cases to a multi-input multi-output control design procedure. The adopted control strategy is finite dimensional and relies on spectral reduction methods. We derive sufficient conditions ensuring the local exponential stabilization of the plant. These control design constraints are shown to be feasible provided the order of the controller is selected to be large enough, ensuring that the reported control design procedure is systematic.

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arXiv2021

Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation

This paper investigates the output feedback setpoint regulation control of a reaction-diffusion equation by means of boundary control. The considered reaction-diffusion plant may be open-loop unstable. The proposed control strategy consists of the coupling of a finite-dimensional observer and a PI controller in order to achieve the boundary setpoint regulation control of various system outputs such as the Dirichlet and Neumann traces. In this context, it is shown that the order of the finite-dimensional observer can always be selected large enough, with explicit criterion, to achieve both the stabilization of the plant and the setpoint regulation of the system output.

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arXiv2021

Global Output Feedback Stabilization of Semilinear Reaction-Diffusion PDEs

This paper addresses the topic of global output feedback stabilization of semilinear reaction-diffusion PDEs. The semilinearity is assumed to be confined into a sector condition. We consider two different types of actuation configurations, namely: bounded control operator and right Robin boundary control. The measurement is selected as a left Dirichlet trace. The control strategy is finite dimensional and is designed based on a linear version of the plant. We derive a set of sufficient conditions ensuring the global exponential stabilization of the semilinear reaction-diffusion PDE. These conditions are shown to be feasible provided the order of the controller is large enough and the size of the sector condition in which the semilinearity is confined into is small enough.

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arXiv2021

Pitchfork-bifurication-based competitive and collaborative control of an E-bike system

This paper is concerned with the design of a human-in-the-loop system for deployment on a smart pedelec (e-bike). From the control-theoretic perspective, the goal is not only to use the power assistance of the e-bike to reject disturbances along the route but also to manage the possibly competitive interactions between a human and the motor intervention. Managing the competitive/cooperative nature of the interactions is crucial for applications in which we wish to control physical aspects of the cycling behavior (e.g. heart rate and breathing rate). The basis of the control is a pitchfork bifurcation system, modeling the interactions, augmented using ideas from gain-scheduling. In vivo experiments have been conducted, showing the effectiveness of the proposed control strategy.

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arXiv2021

Integral action for setpoint regulation control of a reaction-diffusion equation in the presence of a state delay

This paper is concerned with the regulation control of a one-dimensional reaction-diffusion equation in the presence of a state-delay in the reaction term. The objective is to achieve the PI regulation of the right Dirichlet trace with a command selected as the left Dirichlet trace. The control design strategy consists of the design of a PI controller on a finite dimensional truncated model obtained by spectral reduction. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the proposed control design procedure achieves both the exponential stabilization of the original infinite-dimensional system as well as the setpoint regulation of the right Dirichlet trace.

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arXiv2021

Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with a either Dirichlet or Neumann boundary measurement

This paper investigates the output feedback boundary control of reaction-diffusion equations with either distributed or boundary measurement by means of a finite-dimensional observer. A constructive method dealing with the design of finite-dimensional observers for the feedback stabilization of reaction-diffusion equations was reported in a recent paper in the case where either the control or the observation operator is bounded and also satisfies certain regularity assumptions. In this paper, we go beyond by demonstrating that a finite-dimensional state-feedback combined with a finite-dimensional observer can always be successfully designed in order to achieve the Dirichlet boundary stabilization of reaction-diffusion PDEs with a either Dirichlet or Neumann boundary measurement.

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arXiv2021

Boundary Output Feedback Stabilization of Reaction-Diffusion PDEs with Delayed Boundary Measurement

This paper addresses the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs with delayed boundary measurement. The output takes the form of a either Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component which is employed to obtain the control input to be applied at current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided the order of the observer is selected large enough. Taking advantage of this result, we discuss the extension of the control strategy to the case of simultaneous input and output delays.

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arXiv2021

Boundary Output Feedback Stabilization of State Delayed Reaction-Diffusion PDEs

This paper studies the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition while the system output is selected as a either Dirichlet or Neumann boundary trace. The control strategy takes the form of a finite-dimensional observer-based controller with feedback and observer gains that are computed in order to dominate the state delayed term. For any arbitrarily given value of the state delay, we show the exponential stability of the resulting closed-loop system provided the order of the observer is selected large enough.

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arXiv2021

Predictor-Based Output Feedback Stabilization of an Input Delayed Parabolic PDE with Boundary Measurement

This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input delay that the control strategy achieves the exponential stabilization of the closed-loop system, for system trajectories evaluated in $H^1$ norm (also in $L^2$ norm in the case of a Dirichlet boundary measurement), provided the dimension of the observer is selected large enough. The reported proof of this result requires to perform both control design and stability analysis using simultaneously the (non-homogeneous) original version of the PDE and one of its equivalent homogeneous representations.

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