An Introduction to the Control Triple Method for Partial Differential Equations

Schneider, Isabelle

doi:10.1007/978-3-319-64173-7_16

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…As the control terms vanish on the target pattern (u p , v p ), this type of control is called non-invasive. For control of patterns in PDEs, a wide range of possible control terms can be applied, including spatio-temporal delay, proportional feedback, or combinations of these 12,13,21,22 In this paper, we investigate one specific form of non-invasive control, applied to the toy model (2). We leave the u-equation intact, and add proportional feedback control to the v-equation, yielding…”

Section: Controlmentioning

confidence: 99%

Breathing pulses in singularly perturbed reaction-diffusion systems

Veerman

2015

Nonlinearity

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The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.

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Section: Controlmentioning

confidence: 99%

Breathing pulses in singularly perturbed reaction-diffusion systems

Veerman

2015

Nonlinearity

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“…Hence space shifts play an indispensable role for stabilization. Failure of stabilization with pure time delays has also been documented for different models; see [31,32,38]. • As a direct consequence, for the variational case (η, β) = (0, 0) we can selectively stabilize all the unstable vortex equilibria obtained in Lemma 2.3 and also those with the nodal class j = 1 in Lemma 2.4, independently of the number of arms m ∈ N. Our stabilization results, Theorem 3.1 and Theorem 3.2, are novel in the following four aspects.…”

Section: Symmetry-breaking Controls and Main Resultsmentioning

confidence: 99%

“…This article serves as the third episode in which we stabilize certain classes of unstable spiral waves by introducing noninvasive symmetry-breaking feedback controls with spatio-temporal delays. For this purpose we adopt the control triple method introduced by Schneider in [31,32].…”

Section: Introductionmentioning

confidence: 99%

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Schneider¹,

Wolff²,

Dai³

2022

Preprint

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The complex Ginzburg-Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg-Landau m-armed spiral waves have been investigated extensively. However, most spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

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Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

Schneider

Wolff

Dai

2022

Arch Rational Mech Anal

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The complex Ginzburg–Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg–Landau m-armed spiral waves have been investigated extensively. However, many multi-armed spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

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An Introduction to the Control Triple Method for Partial Differential Equations

Cited by 3 publications

References 12 publications

Breathing pulses in singularly perturbed reaction-diffusion systems

Breathing pulses in singularly perturbed reaction-diffusion systems

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

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