### Abstract

The chapter addresses polynomials, for which some remarkable analytical results are available in one special case. It considers the more general case of matrices, focusing on the static output feedback (SOF) problem arising in control of linear dynamical systems. The chapter discusses some spectral radius optimization problems arising in the analysis of the transient behavior of a Markov chain and the design of smooth surfaces using subdivision algorithms. Optimization of roots of polynomials can arise in many contexts, but perhaps the most important application area is feedback control in the frequency domain. As control feedback problems in the frequency domain are a source of applications for polynomial root optimization problems, so control feedback problems in state space are a source of applications for eigenvalue optimization problems. The structure present in a matrix family may lead to local optimizers with active derogatory eigenvalues, even though non-derogatory eigenvalues are the most generic.

Original language | English (US) |
---|---|

Title of host publication | Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations |

Publisher | Wiley Blackwell |

Pages | 351-375 |

Number of pages | 25 |

ISBN (Print) | 9781118577608, 9781848214200 |

DOIs | |

State | Published - Dec 31 2013 |

### Fingerprint

### Keywords

- Eigenvalues
- Markov chain
- Matrices
- Polynomials
- Spectral radius
- Stability optimization
- Static output feedback (SOF)

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations*(pp. 351-375). Wiley Blackwell. https://doi.org/10.1002/9781118577608.ch16

**Stability Optimization for Polynomials and Matrices.** / Overton, Michael L.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations.*Wiley Blackwell, pp. 351-375. https://doi.org/10.1002/9781118577608.ch16

}

TY - CHAP

T1 - Stability Optimization for Polynomials and Matrices

AU - Overton, Michael L.

PY - 2013/12/31

Y1 - 2013/12/31

N2 - The chapter addresses polynomials, for which some remarkable analytical results are available in one special case. It considers the more general case of matrices, focusing on the static output feedback (SOF) problem arising in control of linear dynamical systems. The chapter discusses some spectral radius optimization problems arising in the analysis of the transient behavior of a Markov chain and the design of smooth surfaces using subdivision algorithms. Optimization of roots of polynomials can arise in many contexts, but perhaps the most important application area is feedback control in the frequency domain. As control feedback problems in the frequency domain are a source of applications for polynomial root optimization problems, so control feedback problems in state space are a source of applications for eigenvalue optimization problems. The structure present in a matrix family may lead to local optimizers with active derogatory eigenvalues, even though non-derogatory eigenvalues are the most generic.

AB - The chapter addresses polynomials, for which some remarkable analytical results are available in one special case. It considers the more general case of matrices, focusing on the static output feedback (SOF) problem arising in control of linear dynamical systems. The chapter discusses some spectral radius optimization problems arising in the analysis of the transient behavior of a Markov chain and the design of smooth surfaces using subdivision algorithms. Optimization of roots of polynomials can arise in many contexts, but perhaps the most important application area is feedback control in the frequency domain. As control feedback problems in the frequency domain are a source of applications for polynomial root optimization problems, so control feedback problems in state space are a source of applications for eigenvalue optimization problems. The structure present in a matrix family may lead to local optimizers with active derogatory eigenvalues, even though non-derogatory eigenvalues are the most generic.

KW - Eigenvalues

KW - Markov chain

KW - Matrices

KW - Polynomials

KW - Spectral radius

KW - Stability optimization

KW - Static output feedback (SOF)

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U2 - 10.1002/9781118577608.ch16

DO - 10.1002/9781118577608.ch16

M3 - Chapter

SN - 9781118577608

SN - 9781848214200

SP - 351

EP - 375

BT - Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations

PB - Wiley Blackwell

ER -