arXiv subjects

Cristina Urbani

Publications and source records attributed to Cristina Urbani.

10 recordsLinked to original sources

arXiv2026

Comparison principles and long time behavior for a diffusive Energy Balance Model with vertical resolution

We study a two-layer one-dimensional energy balance model, which allows for vertical energy exchanges between a surface layer and the atmosphere, as well as meridional energy transport across latitudes via a diffusion law. The evolution equations of the surface temperature and the atmospheric temperature are coupled by exchange of infrared radiation as well as other non-radiative energy exchanges. The energy enters the system as solar radiation, which is partially absorbed and partially reflected by the two layers. The system is then composed of two degenerate parabolic equations coupled by nonlinear terms, the growth of these terms being crucial for the choice of the functional setting. An essential parameter is the absorptivity of the atmosphere, denoted $\varepsilon _a$, whose value depends critically on greenhouse gases. We prove that blow up in finite time occurs if $\varepsilon _a >2$, while global existence of solutions and the existence of a global attractor hold when $\varepsilon _a \in (0,2)$. Proofs are based on comparison principles that derive from the cooperative structure of the problem, and that provide invariant rectangles for smooth initial conditions, and on regularity properties.

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arXiv2025

On the small-time bilinear control of a nonlinear heat equation: global approximate controllability and exact controllability to trajectories

In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension $d$. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case $d=1$, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.

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arXiv2023

Analysis of a two-layer energy balance model: long time behaviour and greenhouse effect

We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted $ε_a$. The value of $ε_a$ depends critically on greenhouse gases: increasing concentrations of $CO_2$ and $CH_4$ lead to a more opaque atmosphere with higher values of $ε_a$. First we prove that global existence of solutions of the system holds if and only if $ε_a \in (0, 2)$, and blow up in finite time occurs if $ε_a > 2$. (Note that the physical range of values for $ε_a$ is $(0, 1]$.) Next, we explain the long time dynamics for $ε_a \in (0, 2)$, and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to $ε_a$. This is the key mathematical manifestation of the greenhouse effect.

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arXiv2023

Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker-Planck equation

We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data. The analogous problem (in a more general form) was addressed in our previous paper [Exact controllablity to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker-Planck equation in one space dimension, and a larger class of control actions.

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arXiv2021

Bilinear control of a degenerate hyperbolic equation

We consider the linear degenerate wave equation, on the interval $(0, 1)$ $$ w_{tt} - (x^αw_x)_x = p(t) μ(x) w, $$ with bilinear control $p$ and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the ground state. We prove that, generically with respect to $μ$, any target close to the ground state in the $H^3\times H^2$ topology (suitably adapted to the underlying degenerate operator) is reachable in time $T > \frac{4}{2-α}$, with controls in $L^2((0, T ),\mathbb R)$. Under some classical and generic assumption on $μ$, we prove that there exists a threshold value for time, $T_0= \frac{4}{2-α}$, such that the reachable set is: - a neighborhood of the ground state if $T>T_0$, - contained in a $C^1$-submanifold of infinite codimension if $T<T_0$ - a $C^1$-submanifold of codimension $1$ if $α\in [0,1)$, and a neighborhood of the ground state if $α\in (1,2)$ if $T=T_0$, the case $α=1$ remaining open. This extends to the degenerate case the work [K. Beauchard, Local controllability and non-controllability for a 1D wave equation with bilinear control. J. Differential Equations, 250(4), 2064-2098, 2011] concerning the bilinear control of the classical wave equation ($α=0$), and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering. Control and stabilization of degenerate wave equations. SIAM J. Control Optim., 55(3), 2052-2087, 2017] on the degenerate wave equation where additive control are considered. Our proofs are based on a careful analysis of the spectral problem, and on Ingham type results, which are extensions of the Kadec's $\frac{1}{4}$ theorem.

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arXiv2021

Exact controllability to eigensolutions of the bilinear heat equation on compact networks

Partial differential equation on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control. By adapting a recent method due to [F.~Alabau-Boussouira, P.~Cannarsa and C.~Urbani, {\em Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control}, arXiv:1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.

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arXiv2021

Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-σI$ ($σ\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.

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arXiv2019

Superexponential stabilizability of degenerate parabolic equations via bilinear control

The aim of this paper is to prove the superexponential stabilizability to the ground state solution of a degenerate parabolic equation of the form \begin{equation*} u_t(t,x)+(x^αu_x(t,x))_x+p(t)x^{2-α}u(t,x)=0,\qquad t\geq0,x\in(0,1) \end{equation*} via bilinear control $p\in L_{loc}^2(0,+\infty)$. More precisely, we provide a control function $p$ that steers the solution of the equation, $u$, to the ground state solution in small time with doubly-exponential rate of convergence.\\ The parameter $α$ describes the degeneracy magnitude. In particular, for $α\in[0,1)$ the problem is called weakly degenerate, while for $α\in[1,2)$ strong degeneracy occurs. We are able to prove the aforementioned stabilization property for $α\in [0,3/2)$. The proof relies on the application of an abstract result on rapid stabilizability of parabolic evolution equations by the action of bilinear control. A crucial role is also played by Bessel's functions.

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arXiv2019

Exact controllability to the ground state solution for evolution equations of parabolic type via bilinear control

In a separable Hilbert space $X$, we study the linear evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($p\equiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.

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arXiv2019

Superexponential stabilizability of evolution equations of parabolic type via bilinear control

We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(\cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations.

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