On a class of Riesz-Fischer sequences

Young, Robert M.

doi:10.1090/s0002-9939-98-04416-5

Cited by 6 publications

(6 citation statements)

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“…According to [8], pages 51,80, one can write the heat equation ( 27)- (29) in the form Heat equation ( 32)-( 34) can be written in the semigroup framework (1), where…”

Section: Example Exact Null Controllability Of Interconnected Heat and Delaymentioning

confidence: 99%

“…Some conditions of strong minimality for families of complex exponentials e iλnt , n = 1, 2, ... , where λ n , n = 1, 2, ... are real numbers were discussed in [4,19,29]. Some conditions of strong minimality for families of real exponentials e λnt , n = 1, 2, ... were obtained in [22,23].…”

Section: Example Exact Null Controllability Of Interconnected Heat and Delaymentioning

confidence: 99%

“…. } is said to be strongly minimal (or the Riesz-Fisher sequence [4,19,29,28]), if there exists a positive number γ > 0 such that 8 γ = lim n→∞ γ n , where γ n = inf…”

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confidence: 99%

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

Shklyar¹

2022

DCDS

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<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>

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“…According to [8], pages 51,80, one can write the heat equation ( 27)- (29) in the form Heat equation ( 32)-( 34) can be written in the semigroup framework (1), where…”

Section: Example Exact Null Controllability Of Interconnected Heat and Delaymentioning

confidence: 99%

Section: Example Exact Null Controllability Of Interconnected Heat and Delaymentioning

confidence: 99%

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

Shklyar¹

2022

DCDS

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“…The investigation of the controllability problem defined above is based on the following result of Boas [2] (see also [3] and [18]).…”

Section: Solution Of Moment Problem (24)mentioning

confidence: 99%

“…It may be rather difficult. However if b Y ∈ Y, then one can find more simpler strong minimality conditions for sequence (-exp) in some partial cases (see, for example, results of [1], [21], [22], [11] and references therein) .…”

Section: Partial Controllability Conditionsmentioning

confidence: 99%

On zero controllability of evolution equations by scalar control

Shklyar

2010

Progress in Analysis and Its Applications

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Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Lange

Teismann

2007

Math Methods in App Sciences

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SUMMARYLocal exact controllability of the one-dimensional NLS (subject to zero-boundary conditions) with distributed control is shown to hold in a H 1 -neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. INTRODUCTIONThe control properties of many partial differential equations (PDEs) arising in physics and engineering have been studied extensively. Those investigations include exact and/or optimal controllability for the (linear and nonlinear) heat, wave, beam and plate equations as well as equations of elasticity and Navier-Stokes equations, to mention just some of the most prominent examples. For Schrödinger equations, however, the control theory is markedly less developed. For a brief survey on control results for (linear and nonlinear) Schrödinger equations, see e.g. [1]. The purpose of this paper is to establish local exact controllability for the one-dimensional nonlinear Schrödinger (NLS) equation (subject to zero-boundary conditions) with distributed control. Specifically, we will consider the following control problem: where g (x) denotes the indicator function for some fixed, possibly small, open subinterval ⊂ := (0, 1) representing the spatial region in which the control is applied. Given a fixed control 'horizon' T >0 and initial and target states y 0 and y 1 , the objective is to construct a control function u = u(t, x) that will steer the state y from y 0 to y 1 , i.e. the unique solution y = y(t, x) of (1a)- (1c) is to satisfy (1d). The control problem (1a)-(1d) was posed in [2,3]. A small-data controllability result for periodic boundary conditions is contained in [4].In this paper, we will concentrate on the special case of the focusing cubic NLS equation, i.e. f (s) =−sMore general nonlinearities could be treated, but this is not our main interest here. So we restrict ourselves to the prototypical cubic nonlinearity. Our main result states that the control problem (1a)-(1d) is soluble locally within an H 1 -neighbourhood of the ground-state solution (t, x) = e i t (x), if the control time T >0 is sufficiently large. Here (x) denotes the (nonlinear time-independent) ground state, ‡ i.e. the (real and) positive solution of the boundary value problemwhich is known to exist and to be unique (see Section A.1). Our main result reads. Theorem 1Let ⊂ (0, 1), >0 be given and let denote the ground state. Then, there exist T >0 and >0 such that, for any y 0 ,The theorem will be proved by applying the implicit function theorem (IFT) to the nonlinear map :where t → y(t; y 0 , u) ∈ C([0, T ]; H 1 0 (0, 1)) denotes the unique solution of (1a)-(1c). To be able to apply the IFT, it will be verified that the linearization * u ( , e i T , 0) : Lions [5]. The main difficulty in the application of HUM stems from the lack of self-adjointness of the linear...

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On a class of Riesz-Fischer sequences

Cited by 6 publications

References 6 publications

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

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

On zero controllability of evolution equations by scalar control

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

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