scholarly journals Corrigendum and improvements to “Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations”, and its consequences

2021 ◽  
Vol 272 (1332) ◽  
Author(s):  
Genni Fragnelli ◽  
Dimitri Mugnai
Keyword(s):  
Differential Equations ◽  
Parabolic Equations ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Nonlinear Anal

This paper is a corrigendum of one hypothesis introduced in Mem. Amer. Math. Soc. 242 (2016), no. 1146, and used again in J. Differential Equations 260 (2016), pp. 1314–1371 and Adv. Nonlinear Anal. 6 (2017), pp. 61–84]. We give here the corrected proofs of the concerned results, improving most of them.

scholarly journals Carleman estimates and null controllability of a class of singular parabolic equations

2018 ◽  
Vol 8 (1) ◽  
pp. 1057-1082
Author(s):  
Runmei Du ◽  
Jürgen Eichhorn ◽  
Qiang Liu ◽  
Chunpeng Wang
Keyword(s):  
Control Systems ◽  
Parabolic Equations ◽  
Null Controllability ◽  
Carleman Estimates ◽  
Energy Estimates ◽  
Semilinear Parabolic Equations ◽  
Linearized Equations ◽  
Singular Parabolic Equations ◽  
Semilinear Parabolic

Abstract In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

scholarly journals NULL CONTROLLABILITY OF DEGENERATE NONAUTONOMOUS PARABOLIC EQUATIONS

2019 ◽  
pp. 311
Author(s):  
Abbes Benaissa ◽  
Abdelatif Kainane Mezadek ◽  
Lahcen Maniar
Keyword(s):  
Parabolic Equation ◽  
Open Subset ◽  
Parabolic Equations ◽  
Diffusion Coefficients ◽  
Adjoint System ◽  
Null Controllability ◽  
Carleman Estimates ◽  
One Dimensional ◽  
Neumann Type ◽  
The One

In this paper we are interested in the study of the null controllability for the one dimensional degenerate non autonomous parabolic equation$$u_{t}-M(t)(a(x)u_{x})_{x}=h\chi_{\omega},\qquad  (x,t)\in Q=(0,1)\times(0,T),$$ where $\omega=(x_{1},x_{2})$ is asmall nonempty open subset in $(0,1)$, $h\in L^{2}(\omega\times(0,T))$, the diffusion coefficients $a(\cdot)$ isdegenerate at $x=0$ and $M(\cdot)$ is non degenerate on $[0,T]$. Also the boundary conditions are considered tobe Dirichlet or Neumann type related to the degeneracy rate of $a(\cdot)$. Under some conditions on the functions$a(\cdot)$ and $M(\cdot)$, we prove some global Carleman estimates which will yield the  observability inequalityof the associated adjoint system and equivalently the null controllability of our parabolic equation.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

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.

Smooth conjugacy of centre manifolds

1992 ◽  
Vol 120 (1-2) ◽  
pp. 61-77 ◽  
Author(s):  
Almut Burchard ◽  
Bo Deng ◽  
Kening Lu
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Geometric Theory ◽  
Unstable Manifolds ◽  
Centre Manifolds ◽  
Smooth Conjugacy

SynopsisIn this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

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