arXiv subjects

Christophe Prieur

Publications and source records attributed to Christophe Prieur.

At least 19 recordsLinked to original sources

arXiv2026

H$_1$-ISS Analysis and Boundary Control Design for Coupled Linear ODEs and Hyperbolic PDEs

This paper considers the general problem of the design of boundary controllers for distributed parameter systems. The control objectives are the asymptotic stability of the closed-loop system, as well as the disturbance attenuation of exogenous disturbances affecting the measurements and boundary conditions. More specifically, this work studies input-to-state stability (ISS) and stabilization for systems governed by coupled linear ordinary differential equations (ODEs) and homogeneous linear hyperbolic partial differential equations (PDEs), where actuation, sensing, and disturbance inputs are located at the system boundaries. First, we extend a previously established Lyapunov-based ISS condition for a similar class of systems to the H$_1$ setting under mild assumptions on the disturbance inputs. This extension not only ensures that the H$_1$-norm of the system states driven by non-vanishing disturbances and compatible initial conditions are bounded, but also provides an upper bound on the system H$_1$-norm. Subsequently, these theoretical results are applied to derive stability analysis and control design conditions expressed in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness and potential of the proposed approach.

math.OC↗

arXiv2026

Uniform Exponential Stability Analysis of Impulsive Linear Time-Invariant Systems on Banach and Hilbert Spaces: Non-Coercive and Coercive Stability Conditions

We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are themselves time-invariant, the time-triggered impulses render the propagator a genuinely time-varying evolution family, which is the source of the analysis difficulty addressed here. We combine ideas from hybrid systems theory and infinite-dimensional systems to produce operator-based stability conditions, which can be analytically or numerically checked via convex programming. Necessary and sufficient conditions for the uniform exponential stability of impulsive systems on Banach spaces are obtained in the context of a fixed impulse-times sequence but also of arbitrary, constant, minimum, and range dwell-times using both non-coercive and coercive Lyapunov functionals. Some of those results are then adapted to systems on a Hilbert space and quadratic Lyapunov functionals. As an application, linear switched systems are shown to be an exact special case: reformulated as impulsive systems with unit-norm selector jumps, they inherit non-coercive and clock-dependent dwell-time stability conditions on both Banach and Hilbert spaces. Theoretical and numerical examples are given for illustration, notably on the sampled-data control of time-delay systems.

math.OC↗

arXiv2026

Exponential Decay for a Boundary-Controlled Nonlinear Parabolic Reactor Model

We study an axial dispersion tubular reactor model governed by a nonlinear parabolic equation with Robin-type boundary conditions and boundary feedback control. We derive sufficient conditions for the exponential stability of the steady-state solution of the closed-loop system and provide an explicit estimate of the decay rate. In addition, numerical simulations are presented to illustrate the sharpness of the obtained decay rate for different choices of the feedback gain parameter.

math.AP↗

arXiv2026

Magnetic-Field-Based Localization Using Spatial Field Variations: Signal Processing Principles, Models, and Challenges

Signal processing has played, and continues to play, a fundamental role in the evolution of modern localization technologies. Localization using spatial variations in the Earth's magnetic field is no exception. It relies on signal-processing methods for statistical state inference, magnetic-field modeling, and sensor calibration. Contemporary localization techniques based on spatial variations in the magnetic field can provide decimeter-level indoor localization accuracy and outdoor localization accuracy on par with strategic-grade inertial navigation systems. This article provides a broad, high-level overview of current signal-processing principles and open research challenges in localization using spatial variations in the Earth's magnetic field. The aim is to provide the reader with an understanding of the similarities and differences among existing key technologies from a statistical signal-processing perspective. To that end, existing key technologies will be presented within a common parametric signal-model framework compatible with well-established statistical inference methods.

eess.SP↗

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.

math.OC↗

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.

math.OC↗

arXiv2025

A Unified Representation of Neural Networks Architectures

In this paper we consider the limiting case of neural networks (NNs) architectures when the number of neurons in each hidden layer and the number of hidden layers tend to infinity thus forming a continuum, and we derive approximation errors as a function of the number of neurons and/or hidden layers. Firstly, we consider the case of neural networks with a single hidden layer and we derive an integral infinite width neural representation that generalizes existing continuous neural networks (CNNs) representations. Then we extend this to deep residual CNNs that have a finite number of integral hidden layers and residual connections. Secondly, we revisit the relation between neural ODEs and deep residual NNs and we formalize approximation errors via discretization techniques. Then, we merge these two approaches into a unified homogeneous representation of NNs as a Distributed Parameter neural Network (DiPaNet) and we show that most of the existing finite and infinite-dimensional NNs architectures are related via homogenization/discretization with the DiPaNet representation. Our approach is purely deterministic and applies to general, uniformly continuous matrix weight functions. Relations with neural fields and other neural integro-differential equations are discussed along with further possible generalizations and applications of the DiPaNet framework.

cs.LG↗

arXiv2025

Imitation Learning with Safety and L2 Stability Certificates for Boundary Control of Reaction-Diffusion PDEs

This paper proposes an imitation learning (IL) framework for synthesizing neural network (NN) controllers that achieve boundary stabilization of systems governed by reaction-diffusion partial differential equations (PDEs). The plant is assumed to be actuated through a Dirichlet boundary condition and subject to a Neumann condition on the unactuated side. The design is based on a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system, which is obtained via spectral decomposition. Convex stability and safety conditions are then derived for this truncated model by combining Lyapunov theory with local quadratic constraints (QC), which bound the nonlinear activation functions of the NN and guarantee robustness to model truncation, thus addressing the spillover problem. These conditions are integrated into the IL process to jointly minimize the imitation loss and maximize the volume of the certified region of attraction (ROA). The proposed framework is validated on an unstable reaction-diffusion PDE, demonstrating that the resulting NN controller efficiently reproduces the expert policy while ensuring formal stability guarantees.

math.OC↗

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.

math.OC↗

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.

math.OC↗

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.

math.OC↗

arXiv2025

Revisiting Split Covariance Intersection: Correlated Components and Optimality

Linear fusion is a cornerstone of estimation theory. Implementing optimal linear fusion requires knowledge of the covariance of the vector of errors associated with all the estimators. In distributed or cooperative systems, the cross-covariance terms cannot be computed, and to avoid underestimating the estimation error, conservative fusions must be performed. A conservative fusion provides a fused estimator with a covariance bound that is guaranteed to be larger than the true, but computationally intractable, covariance of the error. Previous research by Reinhardt \textit{et al.} proved that, if no additional assumption is made about the errors of the estimators, the minimal bound for fusing two estimators is given by a fusion called Covariance Intersection (CI). In distributed systems, the estimation errors contain independent and correlated terms induced by the measurement noises and the process noise. In this case, CI is no longer the optimal method. Split Covariance Intersection (SCI) has been developed to take advantage of the uncorrelated components. This paper extends SCI to also take advantage of the correlated components. Then, it is proved that the new fusion provides the optimal conservative fusion bounds for two estimators, generalizing the optimality of CI to a wider class of fusion schemes. The benefits of this extension are demonstrated in simulations.

eess.SP↗

arXiv2024

Cyber-Physical Security of Vehicles: Zero Dynamics Attacks Against Vehicle's Lateral Dynamics

Modern vehicles have evolved from mechanical systems to complex and connected ones controlled by numerous digital computers interconnected through internal networks. While this development has improved their efficiency and safety, it also brings new potential risks, particularly cyber-attacks. Several studies have explored the security of vehicle dynamics against such threats. Among these dynamics, the vehicle's lateral dynamics are crucial for maintaining stability and control during turns and maneuvers, making them a key focus of research. However, only a few recent studies have specifically investigated the security of lateral dynamics. This paper explores the potential for zero dynamics attacks on the vehicle's lateral dynamics, where the attacker can remain undetected by leaving no trace on the system's outputs. Three scenarios are studied: when the output includes yaw rate, lateral acceleration, and their combination. These two critical measurements of a vehicle's lateral motion are accessible through the inertial measurement units (IMU) in every vehicle. For each scenario, the impact of zero dynamics attacks on system performance is analyzed and illustrated through simulations. Finally, the paper provides recommendations for securing vehicles' lateral dynamics against such attacks.

eess.SY↗

arXiv2024

Observer-based output feedback for an age-structured SIRD model

An age-structured Susceptible-Infected-Recovered-Deceased (SIRD) epidemic model is considered. The aim of this paper is to design an observer-based output feedback control law, representing an immunization process, typically vaccination, intended to decrease the peak of infected individuals in the population. At first, well-posedness and stability of the system in open-loop are investigated. Then, to obtain the observer-based output feedback law, a state feedback law is designed by using a normal form. Conditions to ensure stability are established. However, due to physical constraints, this law needs to be adapted. Therefore, a constrained state-feedback law is implemented. This law is designed to fulfill the physical constraints while having good properties (Lipschitz for instance), needed for the last part of the article. Finally, an observer-based output feedback law is obtained using high-gain observer. At each step of the design, convergence properties are obtained. Finally, numerical simulations are performed.

math.OC↗

arXiv2024

Input-to-state stability of infinite-dimensional systems: Foundations and present-day developments

Input-to-state stability (ISS) unifies the stability and robustness in one notion, and serves as a basis for broad areas of nonlinear control theory. In this contribution, we covered the most fundamental facts in the infinite-dimensional ISS theory with a stress on Lyapunov methods. We consider various applications given by different classes of infinite-dimensional systems. Finally, we discuss a Lyapunov-based small-gain theorem for stability analysis of an interconnection of two ISS systems.

eess.SY↗

arXiv2024

Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent Sensing

We study in this paper boundary stabilization, in the L2 sense, of the perturbed Kuramoto-Sivashinsky (KS) equation subject to intermittent sensing. We assume that we measure the state on a given spatial subdomain during certain time intervals, while we measure the state on the remaining spatial subdomain during the remaining time intervals. We assign a feedback law at the boundary of the spatial domain and force to zero the value of the state at the junction of the two subdomains. Throughout the study, the equation's destabilizing coefficient is assumed to be unknown and possibly space dependent but bounded. As a result, adaptive boundary controllers are designed under different assumptions on the perturbation. In particular, we guarantee input-to-state stability (ISS) when an upperbound on the perturbation's size is known. Otherwise, only global uniform ultimate boundedness (GUUB) is guaranteed. In contrast, when the state is measured at every spatial point all the time (full state measurement), convergence to an arbitrarily-small neighborhood of the origin is guaranteed, even if the perturbation's maximal size is unknown. Numerical simulations are performed to illustrate our results.

eess.SY↗

arXiv2024

Split Covariance Intersection with Correlated Components for Distributed Estimation

This paper introduces a new conservative fusion method to exploit the correlated components within the estimation errors. Fusion is the process of combining multiple estimates of a given state to produce a new estimate with a smaller MSE. To perform the optimal linear fusion, the (centralized) covariance associated with the errors of all estimates is required. If it is partially unknown, the optimal fusion cannot be computed. Instead, a solution is to perform a conservative fusion. A conservative fusion provides a gain and a bound on the resulting MSE matrix which guarantees that the error is not underestimated. A well-known conservative fusion is the Covariance Intersection fusion. It has been modified to exploit the uncorrelated components within the errors. In this paper, it is further extended to exploit the correlated components as well. The resulting fusion is integrated into standard distributed algorithms where it allows exploiting the process noise observed by all agents. The improvement is confirmed by simulations.

eess.SP↗

arXiv2024

Improving a Proportional Integral Controller with Reinforcement Learning on a Throttle Valve Benchmark

This paper presents a learning-based control strategy for non-linear throttle valves with an asymmetric hysteresis, leading to a near-optimal controller without requiring any prior knowledge about the environment. We start with a carefully tuned Proportional Integrator (PI) controller and exploit the recent advances in Reinforcement Learning (RL) with Guides to improve the closed-loop behavior by learning from the additional interactions with the valve. We test the proposed control method in various scenarios on three different valves, all highlighting the benefits of combining both PI and RL frameworks to improve control performance in non-linear stochastic systems. In all the experimental test cases, the resulting agent has a better sample efficiency than traditional RL agents and outperforms the PI controller.

eess.SY↗