arXiv subjects

Umberto Biccari

Publications and source records attributed to Umberto Biccari.

At least 37 records · Page 2Linked to original sources

arXiv2021

Control and numerical approximation of fractional diffusion equations

The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain $Ω$. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the $L^2$ and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset $ω\subsetΩ$. Secondly, we move our attention to the exterior control problem, in which the control region $\mathcal O\subsetΩ^c$ is located outside $Ω$. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on $\partialΩ$. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.

math.AP↗

arXiv2021

Multilevel control by duality

We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piece-wise constant control function taking values in a finite-dimensional set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner we build optimal multi-level controls and characterize the time needed for a given ensemble of levels to assure the controllability of the system. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls.

math.OC↗

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.

math.OC↗

arXiv2021

Multiplicity of solutions for fractional $q(.)$-Laplacian equations

In this paper, we deal with the following elliptic type problem $$ \begin{cases} (-Δ)_{q(.)}^{s(.)}u + λVu = α\left\vert u\right\vert^{p(.)-2}u+β\left\vert u\right\vert^{k(.)-2}u & \text{ in }Ω, \\[7pt] u =0 & \text{ in }\mathbb{R}^{n}\backslash Ω, \end{cases} $$ where $q(.):\overlineΩ\times \overlineΩ\rightarrow \mathbb{R}$ is a measurable function and $s(.):\mathbb{R}^n\times \mathbb{R}^n\rightarrow (0,1)$ is a continuous function, $n>q(x,y)s(x,y)$ for all $(x,y)\in Ω\times Ω$, $(-Δ)_{q(.)}^{s(.)}$ is the variable-order fractional Laplace operator, and $V$ is a positive continuous potential. Using the mountain pass category theorem and Ekeland's variational principle, we obtain the existence of a least two different solutions for all $λ>0$. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as $λ\rightarrow +\infty $.

math.AP↗

arXiv2021

Internal control for a non-local Schrödinger equation involving the fractional Laplace operator

We analyze the interior controllability problem for a nonlocal Schrödinger equation involving the fractional Laplace operator $(-Δ)^s$, $s\in(0,1)$, on a bounded $C^{1,1}$ domain $Ω\subset\mathbb{R}^n$. The controllability from a neighborhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. The results follow applying the multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian, and from an explicit computation of the spectrum of the operator in the one-dimensional case.

math.AP↗

arXiv2020

A stochastic approach to the synchronization of coupled oscillators

This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.

nlin.AO↗

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.

math.AP↗

arXiv2019

Controllability properties from the exterior under positivity constraints for a 1-D fractional heat equation

We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator $ (-\partial_x^2)^s$ (with $0<s<1$) on the interval $(-1,1)$. Our control function is localized in an open set $\mathcal O$ in the exterior of $(-1,1)$, that is, $\mathcal O\subset (\mathbb{R} \setminus (-1,1))$. We show that there exists a minimal (strictly positive) time $T_{\rm min}$ such that the fractional heat dynamics can be controlled from any initial datum in $L^2(-1,1)$ to a positive trajectory through the action of an exterior positive control, if and only if $\frac 12<s<1$. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.

math.OC↗

arXiv2019

Controllability of the one-dimensional fractional heat equation under positivity constraints

In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian $(-Δ)^s$ ($0 1/2$. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.

math.AP↗

arXiv2019

Model Reduction of Converter-Dominated Power Systems by Singular Perturbation Theory

The increasing integration of power electronic devices is driving the development of more advanced tools and methods for the modeling, analysis, and control of modern power systems to cope with the different time-scale oscillations. In this paper, we propose a general methodology based on the singular perturbation theory to reduce the order of systems modeled by ordinary differential equations and the computational burden in their simulation. In particular, we apply the proposed methodology to a simplified power system scenario comprised of three inverters in parallel---controlled as synchronverters---connected to an ideal grid. We demonstrate by time-domain simulations that the reduced and decoupled system obtained with the proposed approach accurately represents the dynamics of the original system because it preserves the non-linear dynamics. This shows the efficiency of our technique even for transient perturbations and has relevant applications including the simplification of the Lyapunov stability assessment or the design of non-linear controllers for large-scale power systems.

eess.SY↗

arXiv2019

Dynamics and control for multi-agent networked systems: a finite difference approach

We analyze the dynamics of multi-agent collective behavior models and their control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.

math.AP↗

arXiv2019

Null-controllability properties of a fractional wave equation with a memory term

We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset $ω(t)$ which is moving with a constant velocity $c\in\mathbb{R}$, the main result of the paper states that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.

math.AP↗

arXiv2019

Null-controllability properties of the wave equation with a second order memory term

We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2π\mathbb{Z}$. We assume that the control is acting on an open subset $ω(t)\subset\mathbb{T}$, which is moving with a constant velocity $c\in\mathbb{R}\setminus\{-1,0,1\}$. The main result of the paper shows that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated to our problem and from the application of the classical moment method.

math.AP↗

arXiv2018

WKB expansion for a fractional Schrödinger equation with applications to controllability

This paper is devoted to the analysis of propagation properties for the solutions of a one-dimensional non-local Schrödinger equation involving the fractional Laplace operator $(-d_x^2)^s$, $s\in(0,1)$. We adopt a classical WKB approach and we provide a systematic procedure for building a suitable ansatz for the solutions to the problem. In this way, we can obtain quasi-solutions which are localized along the rays of geometric optics, whose group velocity can be computed explicitly in terms of the parameter $s$. Our results are then confirmed by numerical simulations, based on a finite element approximation of the fractional Laplacian and on a Crank-Nicholson scheme for the time integration. As an application, the controllability problem for the fractional Schrödinger equation is analyzed, finding confirmations of previously known results.

math.AP↗

arXiv2018

Propagation of one and two-dimensional discrete waves under finite difference approximation

We analyze the propagation properties of the numerical versions of one and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and semi-discrete level, by micro-local tools. We do it both for uniform and non-uniform numerical grids and also for constant coefficients and variable ones. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics differ from the continuous to the semi-discrete setting, because of the different nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with the micro-local theory, numerical high-frequency solutions can bend in an unexpected manner, as a result of the accumulation of the local effects introduced by the heterogeneity of the numerical grid. These effects are enhanced in the multi-dimensional case where the interaction and combination of such behaviors in the various space directions may produce, for instance, the "rodeo effect", i. e. waves that are trapped by the numerical grid in closed loops, without ever getting to the exterior boundary. Our analysis allows explaining all such pathological behaviors. Moreover, the discussion in this paper also contributes to the existing theory about the necessity of filtering high-frequency numerical components when dealing with control and inversion problems for waves, which is based very much in the theory of rays and, in particular, on the fact that they can be observed when reaching the exterior boundary of the domain, a key property that can be lost through numerical discretization.

math.AP↗

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.

math.AP↗

arXiv2018

Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential

We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential $$ u_t-u_{xx}-\fracμ{x^2}u=0,\;\;\; (x,t)\in(0,1)\times(0,T).$$ For any $μ<1/4$, we prove that the equation is null controllable through a boundary control $f\in H^1(0,T)$ acting at the singularity point $x=0$. This result is obtained employing the moment method by Fattorini and Russell.

math.AP↗

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.

math.AP↗