scholarly journals Exact null-controllability of interconnected abstract evolution equations with unbounded input operators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Benzion Shklyar
Keyword(s):  
Evolution Equations ◽  
Null Controllability ◽  
Current Paper ◽  
Interconnected Systems ◽  
Controllability Problem ◽  
Abstract Evolution Equations ◽  
Smooth Controls ◽  
Distributed Controls ◽  
Exact Null Controllability ◽  
Bounded Input

<p style='text-indent:20px;'>The exact null-controllability problem in the class of smooth controls with applications to interconnected systems was considered in [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators appearing in systems with distributed controls. The current paper constitutes an extension of the [<xref ref-type="bibr" rid="b23">23</xref>] for the case of unbounded input operators (with more emphasis on the controllability of interconnected systems). The proofs of the results of [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators are adopted for the case of unbounded input operators.</p>

scholarly journals Non null controllability of Stokes equations with memory

2020 ◽  
Vol 26 ◽  
pp. 72
Author(s):  
Enrique Fernández-Cara ◽  
José Lucas F. Machado ◽  
Diego A. Souza
Keyword(s):  
Stokes Equations ◽  
Initial Conditions ◽  
Null Controllability ◽  
Memory Term ◽  
Controllability Problem ◽  
Final Time ◽  
Distributed Controls ◽  
With Memory

In this paper, we consider the null controllability problem for the Stokes equations with a memory term. For any positive final time T > 0, we construct initial conditions such that the null controllability does not hold even if the controls act on the whole boundary. We also prove that this negative result holds for distributed controls.

scholarly journals Local exact controllability of the diffusion equation in one dimension

2003 ◽  
Vol 2003 (14) ◽  
pp. 793-811 ◽  
Author(s):  
Marius Beceanu
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Exact Controllability ◽  
Null Controllability ◽  
One Dimension ◽  
Dirichlet Boundary Value Problem ◽  
Distributed Controls ◽  
Exact Null Controllability

This paper establishes the local exact null controllability of the diffusion equation in one dimension using distributed controls in the case of the Dirichlet boundary value problem. Most of the techniques used in the course of the proof are borrowed from Barbu (2002).

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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.

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