### Abstract

The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance ε of A, We are interested in two aspects of "optimization and pseudoapectra." The first concerns maximizing the function "real part" over an ε-pseudospectrum of a fixed matrix: this defines a function known as the ε-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the ε-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the ε-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small ε; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space - a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.

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

Pages (from-to) | 80-104 |

Number of pages | 25 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - 2004 |

### Fingerprint

### Keywords

- Distance to instability
- Eigenvalue optimization
- H norm
- Nonsmooth analysis
- Pseudospectrum
- Robust control
- Robust optimization
- Spectral abscissa
- Stability radius
- Subdifferential regularity

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*25*(1), 80-104. https://doi.org/10.1137/S0895479802402818

**Optimization and pseudospectra, with applications to robust stability.** / Burke, J. V.; Lewis, A. S.; Overton, M. L.

Research output: Contribution to journal › Article

*SIAM Journal on Matrix Analysis and Applications*, vol. 25, no. 1, pp. 80-104. https://doi.org/10.1137/S0895479802402818

}

TY - JOUR

T1 - Optimization and pseudospectra, with applications to robust stability

AU - Burke, J. V.

AU - Lewis, A. S.

AU - Overton, M. L.

PY - 2004

Y1 - 2004

N2 - The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance ε of A, We are interested in two aspects of "optimization and pseudoapectra." The first concerns maximizing the function "real part" over an ε-pseudospectrum of a fixed matrix: this defines a function known as the ε-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the ε-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the ε-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small ε; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space - a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.

AB - The ε-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance ε of A, We are interested in two aspects of "optimization and pseudoapectra." The first concerns maximizing the function "real part" over an ε-pseudospectrum of a fixed matrix: this defines a function known as the ε-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the ε-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the ε-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small ε; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Hölder continuity property on matrix space - a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.

KW - Distance to instability

KW - Eigenvalue optimization

KW - H norm

KW - Nonsmooth analysis

KW - Pseudospectrum

KW - Robust control

KW - Robust optimization

KW - Spectral abscissa

KW - Stability radius

KW - Subdifferential regularity

UR - http://www.scopus.com/inward/record.url?scp=1342310021&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1342310021&partnerID=8YFLogxK

U2 - 10.1137/S0895479802402818

DO - 10.1137/S0895479802402818

M3 - Article

VL - 25

SP - 80

EP - 104

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 1

ER -