arXiv subjects

Víctor Hernández-Santamaría

Publications and source records attributed to Víctor Hernández-Santamaría.

29 records · Page 2Linked to original sources

arXiv2021

Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

In this paper, we study some controllability and observability problems for stochastic systems coupling fourth- and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on one hand, we study the controllability of a linear backward system where the couplings are made only through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a simplified nonlinear coupled model of forward equations. This case, which is well-known to be harder to solve, follows a methodology that has been introduced recently and relies on an adaptation of the well-known source term method in the stochastic setting together with a truncation procedure. This approach gives a new concept of controllability for stochastic systems.

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arXiv2020

Carleman estimates and controllability results for fully-discrete approximations of 1-D parabolic equations

In this paper, we prove a Carleman estimate for fully-discrete approximations of parabolic operators in which the discrete parameters $h$ and $\triangle t$ are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results for fully-discrete linear and semilinear parabolic equations.

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arXiv2020

Global null-controllability for stochastic semilinear parabolic equations

In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameters are made explicit and are then used in a Banach fixed point method. This allows to circumvent the well-known problem of the lack of compactness embeddings for the solutions spaces arising in the study of controllability problems for stochastic PDEs.

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arXiv2020

Local-controllability of the one-dimensional nonlocal Gray-Scott model with moving controls

In this paper, we prove the local-controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction-diffusion Gray-Scott model when the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and the diffusion coefficient of the second chemical species ${d_v}$ tends to $+ \infty$. The strategy of the proof consists in two main steps. First, we establish the local-controllability of the reaction-diffusion ODE-PDE derived from the Gray-Scott model taking $d_u=0$, and uniformly with respect to the diffusion parameter ${d_v} \in (1, +\infty)$. In order to do this, we prove the (uniform) null-controllability of the linearized system thanks to an observability estimate obtained through adapted Carleman estimates for ODE-PDE. To pass to the nonlinear system, we use a precise inverse mapping argument and, secondly, we apply the shadow limit ${d_v} \rightarrow + \infty$ to reduce to the initial system.

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arXiv2020

Existence and cost of boundary controls for a degenerate/singular parabolic equation

In this paper, we consider the following degenerate/singular parabolic equation $$ u_t -(x^αu_{x})_x - \fracμ{x^{2-α}} u =0, \qquad x\in (0,1), \ t \in (0,T), $$ where $0\leq α<1$ and $μ\leq (1-α)^2/4$ are two real parameters. We prove the boundary null controllability by means of a $H^1(0,T)$ control acting either at $x=1$ or at the point of degeneracy and singularity $x=0$. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters $α$ and $μ$. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.

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arXiv2019

Local null-controllability of a nonlocal semilinear heat equation

This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small-time local null-controllability of a $2 \times 2$ reaction-diffusion system, where the second equation is governed by the parabolic operator $τ\partial_t - σΔ$, $τ, σ> 0$. More precisely, this controllability result is obtained uniformly with respect to the parameters $(τ, σ) \in (0,1) \times (1, + \infty)$. Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit $(τ,σ) \rightarrow (0,+\infty)$. Finally, we illustrate these results by numerical simulations.

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arXiv2019

Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary

In this paper, we present some controllability results for the heat equation in the framework of hierarchic control. We present a Stackelberg strategy combining the concept of controllability with robustness: the main control (the leader) is in charge of a null-controllability objective while a secondary control (the follower) solves a robust control problem, this is, we look for an optimal control in the presence of the worst disturbance. We improve previous results by considering that either the leader or follower control acts on a small part of the boundary. We also present a discussion about the possibility and limitations of placing all the involved controls on the boundary.

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arXiv2018

Null controllability of linear and semilinear nonlocal heat equations with integral kernel

We consider a linear nonlocal heat equation in a bounded domain $Ω\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of null-controllability that allow us to extend the result to some semilinear models.

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arXiv2018

The Poisson equation from non-local to local

We analyze the limit behavior as $s\to 1^-$ of the solution to the fractional Poisson equation $(-Δ)^s u_s=f_s$, $x\inΩ$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0$, $x\inΩ^c$. We show that $\lim_{s\to 1^-} u_s =u$, with $-Δu =f$, $x\inΩ$ and $u=0$, $x\in\partialΩ$. Our results are complemented by a discussion on the rate of convergence and on extensions to the parabolic setting.

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arXiv2017

A Finite Element approximation of the one-dimensional fractional Poisson equation with applications to numerical control

We present a finite element (FE) scheme for the numerical approximation of the solution to a non-local Poisson equation involving the one-dimensional fractional Laplacian $(-d_x^2)^s$ on the interval $(-L,L)$. In particular, we include the complete computations for obtaining the stiffness matrix, starting from the variational formulation of the problem. The problem being one-dimensional, the values of the matrix can be explicitly calculated, without need of any numerical integration, thus obtaining an algorithm which is very efficient in terms of the computational cost. As an application, we analyze the corresponding parabolic equation from the point of view of controllability properties, employing the penalized Hilbert Uniqueness Method (HUM) for computing the numerical approximation of the null-control, acting from an open subset $ω\subset(-L,L)$. In accordance to the theory, our numerical simulations show: (1) that the method solves the elliptic equation with an acceptable approximation in the natural functional setting, (2) that the parabolic problem is null-controllable for $s>1/2$ and (3) that for $s\leq 1/2$ we only have approximate controllability.

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arXiv2016

Robust Stackelberg controllability for a parabolic equation

The aim of this paper is to perform a Stackelberg strategy to control parabolic equations. We have one control, \textit{the leader}, that is responsible for a null controllability property; additionally, we have a control \textit{the follower} that solves a robust control objective. That means, that we seek for a saddle point of a cost functional. In this way, the follower control is not sensitive to a broad class of external disturbances. As far as we know, the idea of combining robustness with a Stackelberg strategy is new in literature

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