Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

Giesl, Peter; Hafstein, Sigurður

doi:10.1016/j.na.2013.03.012

Cited by 28 publications

(33 citation statements)

References 21 publications

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“…Moreover, since the SOS condition is not equivalent to the positivity of a matrix, but just a sufficient condition, one cannot prove a converse theorem, while we will establish such a result in this paper. An algorithm to construct a continuous piecewise affine (CPA) contraction metric for periodic orbits in time-periodic systems using semi-definite optimisation has been proposed in [9].…”

Section: Peter Giesl and Holger Wendlandmentioning

confidence: 99%

Construction of a contraction metric by meshless collocation

Giesl¹,

Wendland²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium. The contraction metric is described by a matrix-valued function M (x) such that M (x) is positive definite and F (M)(x) is negative definite, where F denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation F (M)(x) = −C(x). We then use a construction method based on meshless collocation, developed in the companion paper [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

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Section: Peter Giesl and Holger Wendlandmentioning

confidence: 99%

Construction of a contraction metric by meshless collocation

Giesl¹,

Wendland²

2019

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

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“…Constructive converse theorems, providing algorithms for the explicit construction of a contraction metric, are given in [3] for the global stability of an equilibrium in polynomial systems, using Linear Matrix Inequalities (LMI) and sums of squares (SOS). An algorithm to construct a continuous piecewise affine (CPA) contraction metric for periodic orbits in time-periodic systems using semi-definite optimization has been proposed in [19].…”

Section: Matrix-valued Theorymentioning

confidence: 99%

Kernel-Based Discretization for Solving Matrix-Valued PDEs

Giesl¹,

Wendland²

2018

SIAM J. Numer. Anal.

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In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with a typical example from dynamical systems.

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“…In [11], a contraction metric for a time-periodic ODE with periodic orbit was constructed as a matrix-valued, continuous and piecewise affine (CPA) function. The contraction conditions become the constraints of a semidefinite optimisation problem.…”

Section: Peter Giesl and James Mcmichenmentioning

confidence: 99%

Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation

Giesl¹,

McMichen²

2017

JCD

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A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system. In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a firstorder linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated. In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples.

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Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

Cited by 28 publications

References 21 publications

Construction of a contraction metric by meshless collocation

Construction of a contraction metric by meshless collocation

Kernel-Based Discretization for Solving Matrix-Valued PDEs

Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation

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