On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems

Jacob, Birgit; Nabiullin, Robert; Partington, Jonathan R.; Schwenninger, Felix L.

doi:10.1109/cdc.2016.7798600

Cited by 25 publications

(24 citation statements)

References 18 publications

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“…Thus for this class of systems L ∞ -iISS is equivalent to L ∞ -ISS, and both notions are implied by B ∈ (ℓ q ) −1 , that is, n |bn| q |λn| q < ∞. The following theorem generalizes the main result in [7], where the case q = 2 is studied.…”

Section: Stability Of Parabolic Diagonal Systemsmentioning

confidence: 65%

Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

Jacob

Nabiullin

Partington

et al. 2018

SIAM J. Control Optim.

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103

144

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Abstract. In this work, the relation between input-to-state stability and integral input-to-5 state stability is studied for linear infinite-dimensional systems with an unbounded control operator. 6Although a special focus is laid on the case L ∞ , general function spaces are considered for the inputs. 7We show that integral input-to-state stability can be characterized in terms of input-to-state stability 8 with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated 9 in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions 10 with respect to L ∞ are equivalent. 11Key words. Input-to-state stability, integral input-to-state stability, C 0 -semigroup, admissibil-12 ity, Orlicz spaces 13 AMS subject classifications. 93D20, 93C05, 93C20, 37C75 14

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Section: Stability Of Parabolic Diagonal Systemsmentioning

confidence: 65%

Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

Jacob

Nabiullin

Partington

et al. 2018

SIAM J. Control Optim.

Self Cite

103

144

View full text Add to dashboard Cite

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“…The problems with disturbances acting on the boundaries usually lead to a formulation involving unbounded operators. It is shown in [12], [13] that for a class of linear PDEs, the exponential stability plus a certain admissibility implies the ISS and iISS (integral inputto-state stability [12], [37]) w.r.t. boundary disturbances.…”

Section: Introductionmentioning

confidence: 99%

A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances

Zheng

Zhu

2019

IEEE Trans. Automat. Contr.

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This note addresses input-to-state stability (ISS) properties with respect to (w.r.t.) boundary and in-domain disturbances for Burgers' equation. The developed approach is a combination of the method of De Giorgi iteration and the technique of Lyapunov functionals by adequately splitting the original problem into two subsystems. The ISS properties in L 2 -norm for Burgers' equation have been established using this method. Moreover, as an application of De Giorgi iteration, ISS in L ∞ -norm w.r.t. in-domain disturbances and actuation errors in boundary feedback control for a 1-D linear unstable reaction-diffusion equation have also been established. It is the first time that the method of De Giorgi iteration is introduced in the ISS theory for infinite dimensional systems, and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear partial differential equations (PDEs).

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“…In this paper we develop such criteria for practical ISS for a broad class of infinite-dimensional systems. The understanding of the nature of practical ISS will be beneficial for the development of quantized and sample data controllers for infinite-dimensional systems and will give further insights into the ISS theory of infinite-dimensional systems, which is currently a hot topic [24], [25], [26], [27], [28], [21], [19], [29], [30].…”

Section: Introductionmentioning

confidence: 99%

Criteria for Input-to-State Practical Stability

Mironchenko

2019

IEEE Trans. Automat. Contr.

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For a broad class of infinite-dimensional systems, we characterize input-to-state practical stability (ISpS) using the uniform limit property and in terms of input-to-state stability. We specialize our results to the systems with Lipschitz continuous flows and evolution equations in Banach spaces. Even for the special case of ordinary differential equations our results are novel and improve existing criteria for ISpS.Index Terms-input-to-state stability, nonlinear systems, practical stability, infinite-dimensional systems.

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On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems

Abstract: This is a repository copy of On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems.

Cited by 25 publications

References 18 publications

Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

Infinite-Dimensional Input-to-State Stability and Orlicz Spaces

A De Giorgi Iteration-Based Approach for the Establishment of ISS Properties for Burgers’ Equation With Boundary and In-domain Disturbances

Criteria for Input-to-State Practical Stability

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