null controllability
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Author(s):  
Lijuan Wang ◽  
Can Zhang

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.


Author(s):  
Abdulla Azamov ◽  
Gafurjan Ibragimov ◽  
Khudoyor Mamayusupov ◽  
Marks Ruziboev

AbstractIn this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with $\lambda \in \mathbb {R}$ λ ∈ ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤− 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤− 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered $\ell ^{\infty }$ ℓ ∞ is not asymptotically stable if λ = − 1.


Author(s):  
Brahim Allal ◽  
Genni Fragnelli ◽  
Jawad Salhi

In this paper we study the null controllability for the problems associated to the operators y_t-Ay - \lambda/b(x) y+\int_0^1 K(t,x,\tau)y(t, \tau) d\tau, (t,x) \in (0,T)\times (0,1) where Ay := ay_{xx} or Ay := (ay_x)_x and the functions a and b degenerate at an interior point x0 Ë .0; 1/. To this aim, as a first step we study the well posedness, the Carleman estimates and the null controllability for the associated nonhomogeneous degenerate and singular heat equations. Then,using the Kakutani’s fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.


Author(s):  
Jon Asier Bárcena-Petisco

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time $T$ sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time $T>0$ the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.


2021 ◽  
Author(s):  
Donchyk Yevhenii

Abstract In this paper necessary and sufficient conditions for impulse null-controllability and approximate null-controllability are obtained for the controllability problem for the equation of the string on a rectangle. This article discusses control that is carried out using an external load. Controls solving these problems are found explicitly.


Author(s):  
Long Hu ◽  
Guillaume Olive

The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.


2021 ◽  
Vol 26 (4) ◽  
Author(s):  
Haitham Alghraby ◽  
Radhi A. Zaboon ◽  
Naseif J. Al-Jawari

This paper deals with the problem of controllability and stabilizability of the perturbed linear time-varying nonlocal system defined in some suitable real Hilbert space. The aim of this paper is to show that any globally null-controllable system is completely stabilizability and conversely, under some additional conditions the complete stabilizability implies global null-controllability.


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