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A. Rhandi

Publications and source records attributed to A. Rhandi.

8 recordsLinked to original sources
arXiv2021

Controllability of vertex delay type problems by the regular linear systems approach

In this paper, we study the well-posedness and approximate controllability of a class of network systems having delays and controls at the boundary conditions. The particularity of this work is that the network system is defined on infinite metric graphs. This fact offers many difficulties in applying the usual methods. In fact, the well-posedness of the delay network system is obtained by using a semigroup approach on product spaces which is based on the concept of feedback theory of infinite-dimensional linear systems. This technique allows us to reformulate the delay system into a free-delay distributed control system. From this transformation, we deduce necessary and sufficient conditions for the boundary approximate controllability of such systems. Furthermore, a Rank condition for the approximate controllability is also obtained. This condition coincides with the usual Kalman controllability criterion in the case of a simple transport process on a finite graph. Finally, by applying our approach to a linear Eulerian model (with airborne delays) for (ATFM), we provide a new algebraic condition for the controllability of such a model in terms of the generic rank of the so-called extended controllability matrix.

math.OC↗
arXiv2021

Generation results for vector-valued elliptic operators with unbounded coefficients in L^p spaces

We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,\infty). Sufficient conditions to prove generation results of an analytic C_0-semigroup T(t), together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.

math.AP↗
arXiv2019

Parabolic equations with dynamic boundary conditions and drift terms

The aim of this paper is to study the wellposedness and $L^2$-regularity, firstly for a linear heat equation with dynamic boundary conditions by using the approach of sesquilinear forms, and secondly for its backward adjoint equation using the Galerkin approximation and the extension semigroup to a negative Sobolev space.

math.AP↗
arXiv2018

Vector-valued Schr\"odinger operators on $L^p$-spaces

In this paper we consider vector-valued Schr\"odinger operators of the form $\mathrm{div}(Q\nabla u)-Vu$, where $V=(v_{ij})$ is a nonnegative locally bounded matrix-valued function and $Q$ is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential $V$, we assume an that it is pointwise accretive and that its entries are in $L^\infty_{\mathrm{loc}}(\mathbb{R}^d)$. Under these assumptions, we prove that a realization of the vector-valued Schr\"odinger operator generates a $C_0$-semigroup of contractions in $L^p(\mathbb{R}^d; \mathbb{C}^m)$. Further properties are also investigated.

math.AP↗
arXiv2017

Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ satisfies $$ k(t,x,y) \leq c_1e^{\lambda_0 t+ c_2t^{-\gamma}}\left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(\beta-\alpha)}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}\left(|x|^{\frac{\beta-\alpha+2}{2}}+ |y|^{\frac{\beta-\alpha+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $\lambda_0$ is the largest eigenvalue of the operator $A$, and $\gamma=\frac{\beta-\alpha+2}{\beta+\alpha-2}$, in the case where $N>2,\,\alpha>2$ and $\beta>\alpha -2$. The proof is based on the relationship between the log-Sobolev inequality and the ultracontractivity of a suitable semigroup in a weighted space.

math.AP↗
arXiv2017

Elliptic operators with unbounded diffusion, drift and potential terms

We prove that the realization $A_p$ in $L^p(\mathbb{R}^N),\,1 2,\,\beta >\alpha -2$ and any constants $b\in \mathbb{R}$ and $c>0$. This generalizes the recent results in [A.Canale, A. Rhandi, C. Tacelli, Ann. Sc. Norm. Super. Pisa CI. Sci. (5), 2016] and in [G.Metafune, C.Spina, C.Tacelli, Adv. Diff. Equat., 2014]. Moreover we show that $T(\cdot)$ is consistent, immediately compact and ultracontractive.

math.AP↗
arXiv2014

Analitic approach to solve a degenerate parabolic PDE for the Heston model

We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane.

math.AP↗