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On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Markin Marat V.

Demonstratio Mathematica

2020

Vol. 53, nr 1

352--359

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0 of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C0-semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈R of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L2(R), the anti-self-adjoint differentiation operator A≔ddx with the domain D(A)≔W12(R) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

Controllability of the semilinear Benjamin-Bona-Mahony equation

Leiva H., Merentes N., Sanchez J. L.

Journal of Mathematics and Applications

2012

Vol. 35

39--51

In this paper we prove the interior approximate controllability of the following Generalized Semilinear Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions [formula/wzor] where a ≥ 0 and b > 0 are constans, Ω is a domain in IRN, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0, τ;L2(Ω)) and the nonlinear function ƒ:[0, τ] x IR x IR → IR is smooth enough and there are c, d, e ∈ IR, with c ≠ -1, ea + b > 0 such that [formula/wzor] where Qr = [0, τ] x IR x IR. We prove that for all τ > 0 and any nonempty open subset ω of Ω the system the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the system from an initial state z0 to an ε-neighborhood of the final state z1 on time > 0. As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting a = 0 and b = 1.