Exact Null-Controllability of Abstract Evolution Equations With Boundary Input Operators by Smooth Control

2012 ◽  
Author(s):  
B. Shklyar
Keyword(s):  
Parabolic Equations ◽  
Evolution Equations ◽  
Null Controllability ◽  
Linear Control ◽  
Controllability Problem ◽  
Smooth Control ◽  
Abstract Evolution Equations ◽  
Smooth Controls ◽  
Exact Null Controllability ◽  
Boundary Input

The paper deals with exact null-controllability problem for a linear control system governed by smooth controls. Applications to the exact null-controllability for partial parabolic equations are considered.

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 QUADRATIC DIFFERENTIAL EQUATIONS: PARTIAL GELFAND–SHILOV SMOOTHING EFFECT AND NULL-CONTROLLABILITY

2020 ◽  
pp. 1-53
Author(s):  
Paul Alphonse
Keyword(s):  
Euclidean Space ◽  
Quadratic Differential ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Differential Operators ◽  
Null Controllability ◽  
Geometric Condition ◽  
Positive Time ◽  
Regularizing Effects ◽  
Necessary And Sufficient

We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.

Exact null controllability of degenerate evolution equations with scalar control

2012 ◽  
Vol 203 (12) ◽  
pp. 1817-1836 ◽  
Author(s):  
Vladimir E Fedorov ◽  
Benzion Shklyar
Keyword(s):  
Evolution Equations ◽  
Null Controllability ◽  
Scalar Control ◽  
Exact Null Controllability ◽  
Degenerate Evolution Equations

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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