On parameter-dependent Lyapunov functions for robust stability of linear systems

We investigate several problems in entanglement theory from the perspective of convex optimization. This list of problems comprises (A) the decision whether a state is multi-party entangled, (B) the minimization of expectation values of entanglement witnesses with respect to pure product states, (C) the closely related evaluation of the geometric measure of entanglement to quantify pure multi-party entanglement, (D) the test whether states are multi-party entangled on the basis of witnesses based on second moments and on the basis of linear entropic criteria, and (E) the evaluation of instances of maximal output purities of quantum channels. We show that these problems can be formulated as certain optimization problems: as polynomially constrained problems employing polynomials of degree three or less. We then apply very recently established known methods from the theory of semi-definite relaxations to the formulated optimization problems. By this construction we arrive at a hierarchy of efficiently solvable approximations to the solution, approximating the exact solution as closely as desired, in a way that is asymptotically complete. For example, this results in a hierarchy of novel, efficiently decidable sufficient criteria for multi-particle entanglement, such that every entangled state will necessarily be detected in some step of the hierarchy. Finally, we present numerical examples to demonstrate the practical accessibility of this approach.

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“…conceived as a subset of Ê t , converges (pointwise) to the convex hull of M [12,13,14]. Therefore, we have that…”

Section: A Semi-definite Relaxationsmentioning

confidence: 93%

“…This larger dimension is due to the uplifting procedure used in Lasserre's method [13,14] for approximating the quadratic part of our problems, and goes back to work in Ref. [44].…”

Section: A Semi-definite Relaxationsmentioning

confidence: 99%

Complete hierarchies of efficient approximations to problems in entanglement theory

et al. 2004

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“…[49]. Lastly, asymptotically stability for time-invariant uncertainty of systems like (60) can be investigated through techniques based on positive polynomials also by adopting non-Lyapunov approaches, for example through Hermite's matrices [50] and Hurwitz's determinants [12].…”

Section: Robust Stabilitymentioning

confidence: 99%

LMI Techniques for Optimization Over Polynomials in Control: A Survey

Chesi

2010

IEEE Trans. Automat. Contr.

360

306

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Abstract-Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.

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“…Here, we propose using pointwise maximums of polynomial functions to obtain rich functional forms while keeping the number of optimization decision variables relatively low. Pointwise maximum and other composite Lyapunov functions have been used in many instances, [19], [20], [21], including stability and performance analysis of constrained systems and robustness analysis of uncertain systems, where affine and polynomial parameter-dependent Lyapunov functions are also used, [22], [23]. The notation is generally standard, with R n denoting the set of polynomials with real coefficients in n variables and Σ n ⊂ R n denoting the subset of SOS polynomials.…”

Section: Introductionmentioning

confidence: 99%

Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming

Tan

Packard

2008

IEEE Trans. Automat. Contr.

237

179

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On parameter-dependent Lyapunov functions for robust stability of linear systems

Cited by 61 publications

References 26 publications

Complete hierarchies of efficient approximations to problems in entanglement theory

Complete hierarchies of efficient approximations to problems in entanglement theory

LMI Techniques for Optimization Over Polynomials in Control: A Survey

Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming

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