Stability Optimization for Polynomials and Matrices

Overton, Michael L.

doi:10.1002/9781118577608.ch16

Cited by 9 publications

(11 citation statements)

References 30 publications

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“…We do not have a method to find global minimizers of root max function optimization problems so we approximated them using a local optimization method, as was done in [22] for matrix eigenvalue optimization problems. As explained in [18], the quasi-Newton method known as BFGS, which originated in 1970 to minimize differentiable functions [21], is also extremely effective for finding local minimizers of nonsmooth functions, particularly in the locally Lipschitz case, but also including non-Lipschitz functions such as the root radius, although the same accuracy cannot be expected in the latter case.…”

Section: Resultsmentioning

confidence: 99%

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Polynomial root radius optimization with affine constraints

et al. 2016

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The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most k − 1 inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.

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Section: Resultsmentioning

confidence: 99%

“…Remark 4 If dim null(D L I (0)) = k where D L I is given in (22), then applying Corollary 2 with r = k leads to H ⊆ A, so that any perturbation ∆ ∈ F k is feasible, where A and H are defined in (7) and (17), respectively. Thus, in this case, there exists an optimizer such that all roots are active.…”

Section: Remarkmentioning

confidence: 99%

Polynomial root radius optimization with affine constraints

et al. 2016

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“…Despite the lack of smoothness and convexity of the abscissa and spectral abscissa functionals, numerical methods have been developed that are quite practical for finding local minimizers for arbitrary polynomial and matrix parameterizations-even when these minima correspond to multiple roots or multiple eigenvalues-provided the number of variables and the optimal multiplicities are not too large [38][39][40][41]. It may be interesting to interpret these or related methods in terms of rays and wave fronts propagating in the parameter space.…”

Section: Discussionmentioning

confidence: 99%

Robust stability at the Swallowtail singularity

Kirillov¹,

Overton²

2013

Front. Physics

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Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally minimized at the Swallowtail singular point of the discriminant surface of the set corresponding to a negative real root of multiplicity four. Motivated by this example, we review recent works on robust stability, abscissa optimization, heavily damped systems, dissipation-induced instabilities, and eigenvalue dynamics in order to point out some connections that appear to be not widely known.

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“…We also show that the spectral abscissa is identified to the largest modulus of the zeros of some polynomial defined from the length of this network. For a complete reading on this subject, we can consult Blondel, Gürbüzbalaban, Megretski, and Overton (2012), Kirillov and Overton (2013) and Overton (2014). In these references, we find results that will be useful for the study of more complex situations (see Remark 3.…”

Section: Remarkmentioning

confidence: 95%

Spectral analysis for a degenerate tree and applications

Jellouli

2015

International Journal of Control

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The energy in network of strings subject to the feedback in a root is decreasing to zero if the network is non-degenerate and this decrease is never exponential. We study in this paper the case of degenerate network (i.e. the ratio of two lengths of strings is a rational number) with a feedback in the root of the network. Under this assumption, we determine the energy limit and we identify the best rate of decay with the spectral abscissa of the underlying semi-group generator. We identify the spectral abscissa as in terms of the largest modulus of the roots of a polynomial.

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Stability Optimization for Polynomials and Matrices

Cited by 9 publications

References 30 publications

Polynomial root radius optimization with affine constraints

Polynomial root radius optimization with affine constraints

Robust stability at the Swallowtail singularity

Spectral analysis for a degenerate tree and applications

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