arXiv subjects

Paola Loreti

Publications and source records attributed to Paola Loreti.

At least 37 records · Page 2Linked to original sources

arXiv2020

Generalized binomials in fractional calculus

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities, including an adapted version of the Pascal's rule. We then investigate the associated generating functions, for which we establish a recursive, combinatorial and integral formulation. From this, we derive an asymptotic version of the Binomial Theorem. A combinatorial and asymptotic analysis of some finite sums completes the paper.

math.CO↗

arXiv2020

Fractional SIS epidemic models

In this paper we consider the fractional SIS epidemic model ($α$-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order $α\uparrow 1$ (the SIS model) is also given. We analyse the effects of the fractional derivatives by comparing the SIS and the $α$-SIS models.

math.OC↗

arXiv2020

Time memory effect in entropy decay of Ornstein-Uhlenbeck operators

We investigate the effect of memory terms on the entropy decay of the solutions to equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.

math.AP↗

arXiv2020

Optimal reachability and grasping for a soft manipulator

We investigate optimal reachability and grasping problems for a planar soft manipulator, from both a theoretical and numerical point of view. The underlying control model describes the evolution of the symmetry axis of the device, which is subject to inextensibility and curvature constraints, a bending moment and a curvature control. Optimal control strategies are characterized with tools coming from the optimal control theory of PDEs. We run some numerical tests in order to validate the model and to synthetize optimal control strategies.

math.OC↗

arXiv2019

Alternating and variable controls for the wave equation

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.

math.OC↗

arXiv2018

Modeling and Optimal Control of an Octopus Tentacle

We present a control model for an octopus tentacle, based on the dynamics of an inextensible string with curvature constraints and curvature controls. We derive the equations of motion together with an appropriate set of boundary conditions, and we characterize the corresponding equilibria. The model results in a system of fourth-order evolutive nonlinear controlled PDEs, generalizing the classic Euler's dynamic elastica equation, that we approximate and solve numerically introducing a consistent finite difference scheme. We proceed investigating a reachability optimal control problem associated to our tentacle model. We first focus on the stationary case, by establishing a relation with the celebrated Dubins' car problem. Moreover, we propose an augmented Lagrangian method for its numerical solution. Finally, we address the evolutive case obtaining first order optimality conditions, then we numerically solve the optimality system by means of an adjoint-based gradient descent method.

math.OC↗

arXiv2018

A semilinear integro-differential equation: global existence and hidden regularity

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to state results about global existence of strong and mild solutions without any further smallness on the initial data. Then we define the trace of the normal derivative of the solution showing a regularity result. In such a way we extend to integrodifferential equations with nonlinear term well-known results available in the literature for linear wave equations with memory.

math.AP↗

arXiv2018

Viscoelastic aspects of glass relaxation models

We take advantage of the approximation of the stretched exponential function with a general Prony series in glass relaxation to give some results about the spectral analysis for the equation of viscoelasticity. Moreover, in the case of the Burgers model we carry out a complete investigation that leads to the representation of the solution.

math.AP↗

arXiv2018

Internal observability of the wave equation in a triangular domain

We investigate the internal observability of the wave equation with Dirichlet boundary conditions in a triangular domain. More precisely, the domain taken into exam is the half of the equilateral triangle. Our approach is based on Fourier analysis and on tessellation theory: by means of a suitable tiling of the rectangle, we extend earlier observability results in the rectangle to the case of a triangular domain. The paper includes a general result relating problems in general domains to their tiles, and a discussion of the triangular case. As an application, we provide an estimation of the observation time when the observed domain is composed by three strips with a common side to the edges of the triangle.

math.OC↗

arXiv2018

Variable Support Control for the Wave Equation: A Multiplier Approach

We study the controllability of the multidimensional wave equation in a bounded domain with Dirichlet boundary condition, in which the support of the control is allowed to change over time. The exact controllability is reduced to the proof of the observability inequality, which is proven by a multiplier method. Besides our main results, we present some applications.

math.OC↗

arXiv2018

Modified Fractional Logistic Equation

In the article [B.J.West, Exact solution to fractional logistic equation, Physica A: Statistical Mechanics and its Applications 429 (2015) 103-108], the author has obtained a function as the solution to fractional logistic equation (FLE). As demonstrated later in [I. Area, J. Losada, J. J. Nieto, A note on the fractional logistic equation, Physica A: Statistical Mechanics and its Applications 444 (2016) 182-187], this function (West function) is not the solution to FLE, but nevertheless as shown by West, it is in good agreement with the numerical solution to FLE. The West function indicates a compelling feature, in which the exponentials are substituted by Mittag-Leffler functions. In this paper, a modified fractional logistic equation (MFLE) is introduced, to which the West function is a solution. The proposed fractional integro-differential equation possesses a nonlinear additive term related to the solution of the logistic equation (LE). The method utilized in this article, may be applied to the analysis of solutions to nonlinear fractional differential equations of mathematical physics.

math-ph↗

arXiv2017

Solutions of fractional logistic equations by Euler's numbers

In this paper, we solve in the convergence set, the fractional logistic equation making use of Euler's numbers. To our knowledge, the answer is still an open question. The key point is that the coefficients can be connected with Euler's numbers, and then they can be explicitly given. The constrained of our approach is that the formula is not valid outside the convergence set, The idea of the proof consists to explore some analogies with logistic function and Euler's numbers, and then to generalize them in the fractional case.

math.NT↗

arXiv2017

Determination of order in linear fractional differential equations

In this article, the order of some classes of fractional linear differential equations is determined, based on asymptotic behavior of the solution as time tends to infinity. The order of fractional derivative has been proved to be of great importance in an accurately appropriate simulation of the system under study. Specifically, by representing the asymptotic expansion of the solution, it could be obviously demonstrated that the decay rate of the solution is in influenced by the order of fractional differentiation. The numerical investigation is conducted into the proven formulas.

math.AP↗

arXiv2017

Inverse Observability Inequalities for Integrodifferential Equations in Square Domains

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.

math.AP↗

arXiv2017

Reachability problems for a wave-wave system with a memory term

We solve the reachability problem for a coupled wave-wave system with an integro-differential term. The control functions act on one side of the boundary. The estimates on the time is given in terms of the parameters of the problem and they are explicitly computed thanks to Ingham type results. Nevertheless some restrictions appear in our main results. The Hilbert Uniqueness Method is briefly recalled. Our findings can be applied to concrete examples in viscoelasticity theory.

math.AP↗

arXiv2017

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| α| \le 2} b_α(x,t,η)\partial_x^αu(x,η) dη+ F(x,t) $$ for $x \in Ω$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $Γ\times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $| u(\cdot,0)|_{H^3(Ω)}$ by $\partial_νu|_{Γ\times (0,T)}$ under the assumption that $\partial_tu(\cdot,0) = 0$ and $T>0$ is sufficiently large, $Γ\subset \partialΩ$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.

math.AP↗

arXiv2016

Cantor type functions in non-integer bases

Cantor's ternary function is generalized to arbitrary base-change functions in non-integer bases. Some of them share the curious properties of Cantor's function, while others behave quite differently.

math.CA↗

arXiv2016

Kadec-1/4 Theorem for Sinc Bases

In this paper we show two results. In the first result we consider $λ_n-n=\frac{A}{n^α}$ for $n\in\mathbb N$; if $α>1/2$ and $0<A<\frac{1}{π\sqrt{2 \sqrt{2}ζ(2α)}}$, the system $\left\{\operatorname{sinc}( λ_n - t)\right\}_{n\in\mathbb N}$ is a Riesz basis for $PW_π$. With the second result, we study the stability of $\left\{\operatorname{sinc}( λ_n - t)\right\}_{n\in\mathbb Z}$ for $λ_n\in\mathbb C$; if $|λ_n-n|\leqq L<\frac{1}π\, \sqrt\frac{3α}{8}$, for all $n\in\mathbb Z$, then $\{\operatorname{sinc}(λ_n-t)\}_{n\in\mathbb Z}$ forms a Riesz basis for $PW_π$. Here $α$ is the Lamb-Oseen constant.

math.FA↗