Examinando por Autor "Biccari, Umberto"
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Ítem Dynamics and control for multi-agent networked systems: a finite-difference approach(World Scientific Publishing Co. Pte Ltd, 2019-04) Biccari, Umberto; Ko, Dongnam; Zuazua, EnriqueWe analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal 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.Ítem Internal control for a non-local Schrödinger equation involving the fractional Laplace operator(American Institute of Mathematical Sciences, 2022-02) Biccari, UmbertoWe analyze the interior controllability problem for a non-local Schrödinger equation involving the fractional Laplace operator (-Δ)s, s ϵ(0 1), on a bounded C1 1 domain RN. We first consider the problem in one space dimension and employ spectral techniques to prove that, for s ϵ [1/2/1), null-controllability is achieved through an L2(ωX (0,T)) function acting in a subset ω Ω of the domain. This result is then extended to the multi-dimensional case by applying the classical multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian.Ítem Multilevel Selective Harmonic Modulation via optimal control(Springer, 2022) Biccari, Umberto; Esteve Yagüe, Carlos; Oroya-Villalta, Deyviss JesúsWe consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin’s maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide numerical examples in which the SHM problem is solved by means of our approach.Ítem Null controllability of linear and semilinear nonlocal heat equations with an additive integral kernel(Society for Industrial and Applied Mathematics Publications, 2019) Biccari, Umberto; Hernández Santamaría, VíctorWe consider a linear nonlocal heat equation in a bounded domain Ω ⊂ ℝ with Dirichlet boundary conditions. The nonlocality 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.Ítem Null-controllability properties of the wave equation with a second order memory term(Academic Press Inc., 2019-07-05) Biccari, Umberto; Micu, SorinWe study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus T=R/2πZ. We assume that the control is acting on an open subset ω(t)⊂T, which is moving with a constant velocity c∈R∖{−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 with our problem and from the application of the classical moment method.Ítem Propagation of one- and two-dimensional discrete waves under finite difference approximation(Springer, 2020-12) Biccari, Umberto; Marica, Aurora; Zuazua, EnriqueWe 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 at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable coefficients. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics are different in the continuous and the semi-discrete setting, because of the nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with 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 to explain 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 on 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.Ítem A two-stage numerical approach for the sparse initial source identification of a diffusion–advection equation(Institute of Physics, 2023-09) Biccari, Umberto; Song, Yongcun; Yuan, Xiaoming; Zuazua, EnriqueWe consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion–advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.