Mohamed Camil Belhadjoudja

Mohamed Camil Belhadjoudja

This paper presents a backstepping approach for the boundary control of first-order hyperbolic equations with spatially varying coefficients posed on domains of arbitrary dimension. The method is based on a change of variables induced by the characteristic flow of the time-invariant transport operator, transforming the original multidimensional system into a continuum of decoupled one-dimensional hyperbolic equations evolving along individual characteristic curves. A backstepping controller is then designed for each equation in the decomposition, and the resulting control laws are reassembled in the original coordinates to achieve finite-time stabilization of the full system. The framework relies on the existence of characteristic curves foliating the spatial domain, with uniformly bounded transit times (non-trapping).

Mohamed Camil Belhadjoudja, Mohamed Maghenem, Emmanuel Witrant

We consider a system of two coupled first-order linear hyperbolic partial differential equations modeling heat transport in a counter-flow heat exchanger: one equation describes the transport of a hot fluid, and the other the transport of a cold fluid in the opposite direction. For this system, we design a boundary observer that uses only the temperature of the cold fluid measured at one boundary. Our approach is spectral: by assigning the spectrum of the operator governing the observation error dynamics to a prescribed region within the open left-half complex plane, we can freely tune the convergence rate of the observation error to zero in the $L^2$ norm. The main technical contribution is the proof that spectral stability, that is, the location of the spectrum in the open left-half plane, is equivalent to $L^2$ exponential stability of the origin for the observation error dynamics. This equivalence is established by showing that the operator governing the observation error dynamics satisfies the so-called spectral mapping property.

Mohamed Camil Belhadjoudja, Kirsten A. Morris

We consider the one-dimensional quasilinear heat equation with state-dependent heat capacity and thermal conductivity, and design a boundary-output observer based on the backstepping design for a linear heat equation with constant coefficients. Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the observation error dynamics in $H^1$, with an explicit region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. Importantly, the observation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though, in contrast to backstepping-based stabilization of nonlinear PDEs, the mismatch need not decay along trajectories and may remain bounded away from zero, acting as a persistent state-dependent multiplicative perturbation. A technical challenge was to perform a sufficiently-fine Lyapunov analysis that does not yield overly conservative results such as mere boundedness of the observation error. Interestingly, while in the linear case the relationship between one of the backstepping observer gains and the convergence rate is monotonic, we show that in the nonlinear setting this is no longer the case: there may exist an optimal value of that gain, beyond which further increases deteriorate the system's performance. Such behavior cannot be predicted without our analysis: one might expect a priori the decay rate to be freely tunable at the expense of a region of attraction that shrinks to zero as the prescribed rate tends to infinity. However, our Lyapunov analysis (supported by numerical experiments) reveals that this intuition is incorrect.

Mohamed Camil Belhadjoudja

The goal of this survey paper is to provide an introduction to chaos synchronization using nonlinear observers and its applications in cryptography. I start with an overview of cryptography. Then, I recall the basics of chaos theory and how to use chaotic systems for cryptography, with an introduction to the problem of chaos synchronization. Then, I present the theory of non-linear observers, which is used for the synchronization of chaotic systems. I start with an explanation of the observability problem. Then, I introduce some of the classical observers: Kalman filter, Luenberger observer, Extended Kalman filter, Thau's observer, and High gain observer. I finish by introducing the more advanced observers: Adaptive observers, Unknown inputs observers, Sliding mode observers and ANFIS (Adaptive Neuro-Fuzzy Inference Systems) observers.