arXiv subjects

Hugo Parada

Publications and source records attributed to Hugo Parada.

9 recordsLinked to original sources

arXiv2026

About two results for new valid juggling sequences

In this note, we study two simple operations for extending juggling (siteswap) sequences, called first throws and last catches. Both constructions come from a natural question for a juggler: how can one add throws at the beginning or catches at the end of a pattern without creating a collision? Using landing times and the permutation test, we give necessary and sufficient conditions under which these constructions, when applied to an arbitrary valid siteswap, produce another valid siteswap. We complement our analysis with several examples. An interactive visualization of these extensions is available in [Para].

cs.DM↗

arXiv2026

Bilinear controllability for the linear KdV-Schr{\"o}dinger equation

We study the controllability of a linear KdV-Schr{\"o}dinger equation on the one-dimensional torus via purely imaginary bilinear controls. Considering controls spanning a suitable finite number of Fourier modes, we prove small-time global approximate controllability in L2(T). The result holds between any pair of states with the same norm and is obtained via the saturation method by following the idea introduced in [Poz24]. We first establish small-time controllability for phase multiplications, and then generate transport operators associated with diffeomorphisms of the torus. Finally, we combine these results to recover global approximate controllability. Note that the controllability property holds independently of the Schr{\"o}dinger component of the dynamics, which may in particular be taken to vanish.

eess.SY↗

arXiv2025

Rapid stabilization of the heat equation with localized disturbance

This paper studies the rapid stabilization of a multidimensional heat equation in the presence of an unknown spatially localized disturbance. A novel multivalued feedback control strategy is proposed, which synthesizes the frequency Lyapunov method (introduced by Xiang [41]) with the sign multivalued operator. This methodology connects Lyapunov-based stability analysis with spectral inequalities, while the inclusion of the sign operator ensures robustness against the disturbance. The closed-loop system is governed by a differential inclusion, for which well-posedness is proved via the theory of maximal monotone operators. This approach not only guarantees exponential stabilization but also circumvents the need for explicit disturbance modeling or estimation.

eess.SY↗

arXiv2025

Optimal stabilization rate for the wave equation with hyperbolic boundary condition

We show that the energy of classical solutions to the wave equation with hyperbolic boundary condition (i.e., dynamic Wentzell boundary condition) and damping on the boundary decays like 1/t. In fact we allow mixed boundary conditions: a possibly empty, disjoint part of the boundary may be kept at rest provided that the dynamic part satisfies the geometric control condition. We also prove that this decay rate is sharp. Our results follow from resolvent estimates, which we establish by studying high-frequency quasimodes.

math.AP↗

arXiv2025

A Nonhomogeneous Boundary-Value Problem For The Nonlinear KdV Equation on Star Graphs

This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of $s$-compatibility, which generalizes classical compatibility conditions to star-shaped and more complex graph configurations, inspired by the works of Bona, Sun, and Zhang [14]. By combining analytical techniques with a fixed-point argument, we establish sharp global well-posedness for both the linear and nonlinear problems at the $H^s$ level. In this setting, our results extend the classical analysis for a single KdV equation [14] to star-shaped graphs composed of $N$ equations. These results provide the first comprehensive well-posedness theory for KdV equations with coupled boundary conditions on graphs. Although control issues are not treated in this article, the analytic results obtained here address several open problems, which will be addressed in a forthcoming

math.AP↗

arXiv2025

Feedback stabilization of some fourth-order nonlinear parabolic equations with saturated controls

In this work, we analyze the internal and boundary stabilization of the Cahn-Hilliard and Kuramoto-Sivashinsky equations under saturated feedback control. We conduct our study through the spectral analysis of the associated linear operator. We identify a finite number of eigenvalues related to the unstable part of the system and then design a stabilization strategy based on modal decomposition, linear matrix inequalities (LMIs), and geometric conditions on the saturation function. Local exponential stabilization in $H^{2}$ is established.

eess.SY↗

arXiv2025

Rapid stabilization for a wave equation with boundary disturbance

In this paper, we study the rapid stabilization of an unstable wave equation, in which an unknown disturbance is located at the boundary condition. We address two different boundary conditions: Dirichlet- Dirichlet and Dirichlet-Neumann. In both cases, we design a feedback law, located at the same place as the unknown disturbance, that forces the exponential decay of the energy for any desired decay rate while suppressing the effects of the unknown disturbance. For the feedback design, we employ the backstepping method, Lyapunov techniques and the sign multivalued operator. The well-posedness of the closed-loop system, which is a differential inclusion, is shown with the maximal monotone operator theory.

math.OC↗

arXiv2025

Spectral scheme for an energetic Fokker-Planck equation with $\kappa$-distribution steady states

The concern of the present paper is the design of efficient numerical schemes for a specific Fokker-Planck equation describing the dynamics of energetic particles occurring in thermonuclear fusion plasmas (runaway electrons for example). In the long-time limit, the velocity distribution function of these particles tends towards a thermal non-equilibrium $\kappa$-distribution function which is a steady-state of the considered Fokker-Planck equation. These $\kappa$-distribution functions have the particularity of being only algebraically decaying for large velocities, thus describing very well suprathermal particle populations. Our aim is to present two efficient spectral methods for the simulation of such energetic particle dynamics. The first method will be based on rational Chebyshev basis functions, rather than on Hermite basis sets, which are the basis of choice for Maxwellian steady states. The second method is based on a different polynomial basis set, constructed via the Gram-Schmidt orthogonalisation process. These two new spectral schemes, specifically adapted to the here considered physical context, shall permit to cope with the long-time asymptotics without significant numerical costs.

math.AP↗

arXiv2025

An analysis of the entire functions associated with the operator of the KdV equation

It is well known that the controllability property of partial differential equations (PDEs) is closely linked to the proof of an observability inequality for the adjoint system, which, sometimes, involves analyzing a spectral problem associated with the PDE under consideration. In this work, we study a series of spectral issues that ensure the controllability of the renowned Korteweg-de Vries equation on a star-graph. This investigation reduces to determining when certain functions, associated with this spectral problem, are entire. The novelty here lies in presenting this detailed analysis in the context of a star graph structure.

math.AP↗