Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

Ahuja, Sunil; Kevrekidis, Ioannis G.; Rowley, Clarence W.

doi:10.1007/s00332-005-0801-7

Cited by 6 publications

(8 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are a number of simulation studies on designing a state feedback control for the KS equation; see for instance, [1], [18], [22], [23], [30]. In [23] both linear and nonlinear feedback controls were designed to stabilize the KS equation.…”

Section: Introductionmentioning

confidence: 99%

“…The basic idea is to choose the boundary conditions so that the energy of the nonlinear system decays to zero exponentially resulting exponential stability of the zero equilibrium. Another approach that is used to stabilize the KS equation via distributed control is by stabilizing the corresponding linearized system, then use the same controller to stabilize the nonlinear KS equation [1], [18], [3], [4], [9], [22], [23], [26], [30].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Output feedback control of the Kuramoto-Sivashinsky equation

Jamal

Morris

2015

2015 54th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

The Kuramoto-Sivashinsky equation is a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibria depends on the value of a positive parameter; the set of all constant equilibria are unstable when the instability parameter is less than 1. Stabilization of the Kuramoto-Sivashinsky equation using scalar output-feedback control is considered in this paper. This is done by stabilizing the corresponding linearized system. A finite-dimensional controller is then designed to stabilize the system. Fréchet differentiability of the semigroup generated by the closed-loop system plays an important role in proving that this approach yields a locally stable equilibrium. The approach is illustrated with a numerical example.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Output feedback control of the Kuramoto-Sivashinsky equation

Jamal

Morris

2015

2015 54th IEEE Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

show abstract

“…Normally, this global alignment (the computation of the symmetry group element identified with each snapshot) may be accomplished numerically through the use of a wellchosen template function (see Figure 1 and, e.g., Ahuja et al (2007); Rowley & Marsden (2000)). For instance, in our running example of snapshots {u(θ, t k )} M k=1 , one finds the alignments {θ k } M k=1 which align each snapshot with a template T (θ) by simply setting…”

Section: Time Positionmentioning

confidence: 99%

Noisy dynamic simulations in the presence of symmetry: Data alignment and model reduction

Sonday

Singer

Kevrekidis

2013

Computers & Mathematics with Applications

View full text Add to dashboard Cite

We process snapshots of trajectories of evolution equations with intrinsic symmetries, and demonstrate the use of recently developed eigenvector-based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise.Our illustrative examples include a one-dimensional evolutionary partial differential (the Kuramoto-Sivashinsky) equation with periodic boundary conditions, as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear Smoluchowski equation on the surface of a sphere. This is a useful first step towards data mining the "symmetry-adjusted" ensemble of snapshots in search of an accurate low-dimensional parametrization (and the associated reduction of the original dynamical system). We also demonstrate a technique ("vector diffusion maps") that combines, in a single formulation, the symmetry removal step and the dimensionality reduction step.

show abstract

“…The basic idea is to choose the boundary conditions so that the energy of the nonlinear system decays to zero exponentially. Distributed control of the KS equation has been approached by stabilizing the corresponding linearized system [1,4,33,37,40,49].…”

Section: Introduction the Kuramoto-sivashinsky (Ks) Equation Was Intmentioning

confidence: 99%

Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation

Jamal¹,

Morris²

2018

SIAM J. Control Optim.

View full text Add to dashboard Cite

Abstract. Linearization is a useful tool for analyzing the stability of nonlinear differential equations. Unfortunately, the proof of the validity of this approach for ordinary differential equations does not generalize to all nonlinear partial differential equations. General results giving conditions for when stability (or instability) of the linearized equation implies the same for the nonlinear equation are given here. These results are applied to stability and stabilization of the Kuramoto-Sivashinsky equation, a nonlinear partial differential equation that models reaction-diffusion systems. The stability of the equilibrium solutions depends on the value of a positive parameter ν. It is shown that if ν > 1, then the set of constant equilibrium solutions is globally asymptotically stable. If ν < 1 then the equilibria are unstable. It is also shown that stabilizing the linearized equation implies local exponential stability of the equation. Stabilization of the Kuramoto-Sivashinsky equation using a single distributed control is considered and it is described how to use a finite-dimensional approximation to construct a stabilizing controller. The results are illustrated with simulations.

show abstract

Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

Cited by 6 publications

References 35 publications

Output feedback control of the Kuramoto-Sivashinsky equation

Output feedback control of the Kuramoto-Sivashinsky equation

Noisy dynamic simulations in the presence of symmetry: Data alignment and model reduction

Linearized Stability of Partial Differential Equations with Application to Stabilization of the Kuramoto--Sivashinsky Equation

Contact Info

Product

Resources

About