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Víctor Hernández-Santamaría

Publications and source records attributed to Víctor Hernández-Santamaría.

At least 19 recordsLinked to original sources
arXiv2026

Stability for the stochastic heat equation with multiplicative noise via finite-dimensional feedback

In this paper, we study the long-time behavior of a stochastic heat equation with a class of infinite-dimensional multiplicative noises and localized control. We first illustrate through simple examples of the uncontrolled dynamics that mean-square and almost sure exponential stability may hold under different conditions on the parameters, reflecting the interplay between the drift and the multiplicative noise. We then construct a finite-dimensional feedback control acting on a measurable subset of positive measure, built from finitely many Fourier modes of the solution. In particular, we show that the number of controlled modes determines the decay rate and allows for arbitrarily fast stabilization in the mean-square sense. As a consequence, almost sure exponential stability is recovered via a probabilistic argument, so that both notions of stability are achieved within the same framework and with the same guaranteed decay rate. We also establish null-controllability at any positive time through an iterative construction of adapted controls from finite-dimensional feedback laws. The construction relies on the quantitative stabilization estimates and requires no observability inequality for the adjoint equation.

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arXiv2026

A Lebeau--Robbiano approach to the controllability of the Boussinesq system with a reduced number of controls

In this paper, we prove the local null controllability of the Boussinesq system in dimensions two and three with a reduced number of localized controls. More precisely, the controls act on the temperature equation and on only $N-2$ components of the velocity equation. Our proof is based on a spectral approach in the spirit of the Lebeau--Robbiano method. The main difficulty comes from the coupled structure of the system. The velocity and temperature equations are governed by the Stokes operator and the Dirichlet Laplacian, respectively. Their natural spectral decompositions are different, and there is no common spectral localization naturally adapted to the coupled system. Therefore, the usual componentwise spectral argument cannot be applied directly. We show that the cascade structure of the linearized system makes it possible to overcome this difficulty. The main ingredient is a mixed observability estimate in which only the Stokes component is spectrally localized, while the heat component is treated without any frequency restriction. Combining this estimate with the dissipation of the high Stokes frequencies allows us to carry out a Lebeau--Robbiano iteration for the linearized system. The resulting linear controllability estimate is transferred to the nonlinear Boussinesq system through a time-iteration argument. As an important consequence, the controls have a small-time cost bounded by $C\exp(C/T)$, which recovers the expected parabolic order while preserving the reduced number of controls.

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arXiv2026

$L^{p}$-estimates, local well-posedness and controllability for linear and semilinear backward SPDEs

In this paper, we study linear backward parabolic SPDEs in bounded domains and present new a priori estimates for their weak solutions. Inspired by the seminal work of Y. Hu, J. Ma and J. Yong from 2002 on strong solutions, we establish $L^p$-estimates requiring minimal assumptions on the regularity of the coefficients, the terminal data, and the external force. Our approach relies on direct, constructive, and quantitative arguments, adapted from known methods in the theory of SPDEs to this setting. In particular, we develop a new Itô's formula for the $L^p$-norm of the backward solution, tailored to this setting and extending the classical result in the $L^2$-framework. This formula is then used to improve further the regularity of the first component of the solution up to $L^\infty$. We also present two applications: a local existence result for a semilinear equation without imposing any growth condition on the nonlinear term, and a novel local controllability result for semilinear backward SPDEs that partially resolves an open problem in the field.

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arXiv2026

Global controllability of the Cahn-Hilliard equation

This paper investigates the global controllability properties of the Cahn--Hilliard equation posed on the $d$-dimensional flat torus $\mathbb{T}^d$. We first establish small-time global approximate controllability of the system by means of controls acting on finitely many Fourier modes, relying on techniques inspired by geometric control theory. We then prove null controllability of the linearized equation using a spatially localized control supported on an arbitrary measurable subset of positive Lebesgue measure, based on quantitative propagation of smallness estimates for the free dynamics. For dimensions $d \in \{1,2,3\}$, we further derive local null controllability for the full nonlinear system via a fixed-point argument. By combining these results, we establish global null controllability of the Cahn--Hilliard equation. This work provides the first result on global controllability for this equation, achieved through a two-stage strategy in which the control is first localized in Fourier space and subsequently restricted to a set of positive measure.

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arXiv2025

On the controllability of the Kuramoto-Sivashinsky equation on multi-dimensional cylindrical domains

In this article, we investigate null controllability of the Kuramoto-Sivashinsky (KS) equation on a cylindrical domain $Ω=Ω_x\times Ω_y$ in $\mathbb R^N$, where $Ω_x=(0,a),$ $a>0$ and $Ω_y$ is a smooth domain in $\mathbb R^{N-1}$. We first study the controllability of this system by a control acting on $\{0\}\times ω$, $ω\subset Ω_y$, through the boundary term associated with the Laplacian component. The null controllability of the linearized system is proved using a combination of two techniques: the method of moments and Lebeau-Robbiano strategy. We provide a necessary and sufficient condition for the null controllability of this system along with an explicit control cost estimate. Furthermore, we show that there exists minimal time $T_0(x_0)>0$ such that the system is null controllable for all time $T > T_0(x_0)$ by means of an interior control exerted on $γ= \{x_0\} \times ω\subset Ω$, where $x_0/a\in (0,1)\setminus \mathbb{Q}$ and it is not controllable if $T 1$, then we prove the controllability for any time $T>0.$ Finally, for the case of $N=2 \text{ or } 3$, we show the local null controllability of the main nonlinear system by employing the source term method followed by the Banach fixed point theorem.

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arXiv2025

Boundary null controllability of a class of 2-d degenerate parabolic PDEs

This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof combines two classical techniques: the method of moments and the Lebeau-Robbiano strategy. A key novelty of this work lies in the analysis of boundary control localized on a subset of the boundary where the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to $N$-dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.

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arXiv2025

FEM for 1D-problems involving the logarithmic Laplacian: error estimates and numerical implementation

We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error estimates and provide a logarithmic order of convergence in the energy norm using suitable $\log$-weighted spaces. Moreover, we show that the stiffness matrix of logarithmic problems can be obtained as the derivative of the fractional stiffness matrix evaluated at $s=0$. Lastly, we investigate the relationship between the discrete eigenvalue problem and its convergence to the continuous one.

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arXiv2024

Event-triggered boundary control of the linearized FitzHugh-Nagumo equation

In this paper, we address the exponential stabilization of the linearized FitzHugh-Nagumo system using an event-triggered boundary control strategy. Employing the backstepping method, we derive a feedback control law that updates based on specific triggering rules while ensuring the exponential stability of the closed-loop system. We establish the well-posedness of the system and analyze its input-to-state stability in relation to the deviations introduced by the event-triggered control. Numerical simulations demonstrate the effectiveness of this approach, showing that it stabilizes the system with fewer control updates compared to continuous feedback strategies while maintaining similar stabilization performance.

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arXiv2024

Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction

We study the existence and concentration of positive and nodal solutions to a Schrödinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting profile that solves a nonautonomous elliptic semilinear equation with a sharp sign change in the nonlinearity. We characterize the (radial or foliated Schwarz) symmetries and the (polynomial) decay of the least-energy positive and nodal limiting profiles.

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arXiv2023

Controllability issues for parabolic-elliptic systems involving nonlocal couplings

This work addresses controllability properties for some systems of partial differential equations in which the main feature is the coupling through nonlocal integral terms. In the first part, we study a nonlinear parabolic-elliptic system arising in mathematical biology and, using recently developed techniques, we show how Carleman estimates can be directly used to handle the nonlocal terms, allowing us to implement well-known strategies for controlling coupled systems and nonlinear problems. In the second part, we investigate fine controllability properties of a 1-d linear nonlocal parabolic-parabolic system. In this case, we will see that the controllability of the model can fail and it will depend on particular choices and combinations of local and nonlocal couplings.

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arXiv2023

Controllability of some semilinear shadow reaction-diffusion systems

The shadow limit is a versatile tool used to study the reduction of reaction-diffusion systems into simpler PDE-ODE models by letting one of the diffusion coefficients tend to infinity. This reduction has been used to understand different qualitative properties and their interplay between the original model and its reduced version. The aim of this work is to extend previous results about the controllability of linear reaction-diffusion equations and how this property is inherited by the corresponding shadow model. Defining a suitable class of nonlinearities and improving some uniform Carleman estimates, we extend the results to the semilinear case and prove that the original model is null-controllable and that the shadow limit preserves this important feature.

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arXiv2023

Insensitizing control problems for the stabilized Kuramoto-Sivashinsky system

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for controls such that some functional of the state (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying a local inversion theorem.

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arXiv2022

Small order asymptotics for nonlinear fractional problems

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order $2s$ when the parameter $s$ tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e., the pseudodifferential operator with Fourier symbol $\ln(|ξ|^2)$. These results are motivated by some applications of nonlocal models where a small value for the parameter $s$ yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.

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arXiv2021

Optimal control of linear non-local parabolic problems with an integral kernel

We consider a linear non-local heat equation in a bounded domain $Ω\subset\mathbb{R}^d$, $d\geq 1$, with Dirichlet boundary conditions, where the non-locality is given by the presence of an integral kernel. Motivated by several applications in biological systems, in the present paper we study some optimal control problems from a theoretical and numerical point of view. In particular, we will employ the classical low-regret approach of J.-L. Lions for treating the problem of incomplete data and provide a simple computational implementation of the method. The effectiveness of the results are illustrated by several examples.

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arXiv2021

Existence and convergence of solutions to fractional pure critical exponent problems

We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical problem \begin{equation*} (-Δ)^su_s=|u_s|^{2^\star_s-2}u_s, \quad u_s\in D^s_0(Ω),\quad 2^\star_s:=\frac{2N}{N-2s}, \end{equation*} where $s$ is any positive number, $Ω$ is either $\mathbb{R}^N$ or a smooth symmetric bounded domain, and $D^s_0(Ω)$ is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign changing. We show that, up to a subsequence, a l.e.s.s. $u_s$ converges to a l.e.s.s. $u_{t}$ as $s$ goes to any $t>0$. In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order $t-\varepsilon$. A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If $t$ is an integer, these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any $s>1$.

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arXiv2021

Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $ϕ(\triangle t)$-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $\triangle t$ goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.

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