scholarly journals Identification of a convolution kernel in a control problem for the heat equation with a boundary memory term

2012 ◽  
Vol 193 (3) ◽  
pp. 779-816 ◽  
Author(s):  
Cecilia Cavaterra ◽  
Davide Guidetti
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Memory Term ◽  
Convolution Kernel

scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals Insensitizing control for linear and semi-linear heat equations with partially unknown domain

2019 ◽  
Vol 25 ◽  
pp. 50 ◽  
Author(s):  
Pierre Lissy ◽  
Yannick Privat ◽  
Yacouba Simporé
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Duality Theory ◽  
Unique Continuation ◽  
Dirichlet Boundary Conditions ◽  
Linear Case ◽  
Heat Equations ◽  
Geometrical Approach ◽  
Linear Heat Equation ◽  
Linear Heat

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

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