scholarly journals Local null controllability of the penalized Boussinesq system with a reduced number of controls

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jon Asier Bárcena-Petisco ◽  
Kévin Le Balc'h
Keyword(s):  
Adjoint System ◽  
Null Controllability ◽  
Dirichlet Boundary Conditions ◽  
Boussinesq System ◽  
Order Differential Operator ◽  
Control Cost ◽  
Incompressibility Condition ◽  
Open Set ◽  
Linearized System ◽  
Homogeneous Dirichlet Boundary Conditions

<p style='text-indent:20px;'>In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb R^N $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ N = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon &gt; 0 $\end{document}</tex-math></inline-formula>. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set <inline-formula><tex-math id="M5">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> contained in <inline-formula><tex-math id="M6">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. We also show that the control cost is bounded uniformly with respect to <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.</p>

scholarly journals Null controllability of a nonlinear population dynamics problem

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Oumar Traore
Keyword(s):  
Population Dynamics ◽  
Internal Control ◽  
Initial Density ◽  
Adjoint System ◽  
Null Controllability ◽  
Dynamics Model ◽  
Carleman Inequality ◽  
Recruitment Process ◽  
Open Set ◽  
Population Dynamics Model

We establish a null controllability result for a nonlinear population dynamics model. In our model, the birth term is nonlocal and describes the recruitment process in newborn individuals population. Using a derivation of Leray-Schauder fixed point theorem and Carleman inequality for the adjoint system, we show that for all given initial density, there exists an internal control acting on a small open set of the domain and leading the population to extinction.

scholarly journals Local null controllability of a model system for strong interaction between internal solitary waves

2021 ◽  
pp. 2150003
Author(s):  
Jon Asier Bárcena-Petisco ◽  
Sergio Guerrero ◽  
Ademir F. Pazoto
Keyword(s):  
Source Term ◽  
Coupled System ◽  
Adjoint System ◽  
Internal Gravity Waves ◽  
Null Controllability ◽  
Bounded Interval ◽  
Nonlinear Coupled System ◽  
Local Inversion ◽  
Linearized System ◽  
Controllability Property

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

scholarly journals Null controllability of a nonlinear heat equation

2002 ◽  
Vol 7 (7) ◽  
pp. 375-383 ◽  
Author(s):  
G. Aniculăesei ◽  
S. Aniţa
Keyword(s):  
Heat Equation ◽  
Dirichlet Boundary ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Nonlinear Heat Equation ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Linearized System ◽  
Kakutani Fixed Point ◽  
Exact Null Controllability ◽  
Kakutani Fixed Point Theorem

We study the internal exact null controllability of a nonlinear heat equation with homogeneous Dirichlet boundary condition. The method used combines the Kakutani fixed-point theorem and the Carleman estimates for the backward adjoint linearized system. The result extends to the case of boundary control.

scholarly journals Scalar Casimir effect for a conducting cylinder in a Lorentz-violating background

2021 ◽  
pp. 2150168
Author(s):  
A. M. Escobar-Ruiz ◽  
A. Martín-Ruiz ◽  
C. A. Escobar ◽  
Román Linares
Keyword(s):  
Cylindrical Shell ◽  
Green's Functions ◽  
Green’S Functions ◽  
Casimir Effect ◽  
Normal Component ◽  
Vacuum Expectation ◽  
Dirichlet Boundary Conditions ◽  
Order Differential Operator ◽  
Energy Tensor ◽  
Approximate Analytical Expression

Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green’s functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal–normal component of the stress–energy tensor) as a suitable second-order differential operator acting on the corresponding Green’s functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force.

scholarly journals The lower bound of number of solutions for the second order nonlinear boundary value problem via the root functions method

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Peter Somora
Keyword(s):  
Lower Bound ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Root Functions ◽  
Number Of Zeros ◽  
Variational Solutions ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractWe consider a second order nonlinear differential equation with homogeneous Dirichlet boundary conditions. Using the root functions method we prove a relation between the number of zeros of some variational solutions and the number of solutions of our boundary value problem which follows into a lower bound of the number of its solutions.

scholarly journals Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Anyin Xia ◽  
Xianxiang Pu ◽  
Shan Li
Keyword(s):  
Porous Medium ◽  
Global Regularity ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Singularity Analysis ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Medium Equation ◽  
Regularity Results ◽  
Homogeneous Dirichlet Boundary Conditions

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

A Lane–Emden system with singular data

2011 ◽  
Vol 141 (6) ◽  
pp. 1279-1294 ◽  
Author(s):  
Marius Ghergu
Keyword(s):  
Boundary Conditions ◽  
Elliptic System ◽  
Bounded Domain ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Singular Data

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.

Nonlinear Dirichlet problem with non local regional diffusion

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
César E. Torres Ledesma
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Dirichlet Problem ◽  
Non Local ◽  
Homogeneous Dirichlet Boundary Conditions

AbstractThe purpose of this paper is to study the existence of solutions for equations driven by a non-local regional operator with homogeneous Dirichlet boundary conditions. More precisely, we consider the problemwhere the nonlinear term

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