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Title:听Robust stability of linear systems of linear delay differential equations.

础产蝉迟谤补肠迟:听

We consider a class of nonlinear eigenvalue problems including those arising from stability analysis of linear systems of delay differential equations.Our aim is to compute the pseudospectral abscissa, i.e. the real part of the rightmost point in the pseudospectrum, and detect - for a stable system - when the pseudospectral abscissa vanishes, which听determines a nearby system which is not anymore stable. If this occurs for a small perturbation of the considered stable system, this is a sign of lack of robustness.

In analogy to the linear eigenvalue problem we have that it is sufficient to restrict the analysis听to rank-1 perturbations of the matrices of the system.听Using this main result we present a gradient system approach which only requires the computation of听the spectral abscissa of a sequence of problems obtained by adding rank one updates to the matrices.听In order to be applied these methods simply require a procedure to compute the rightmost eigenvalue听and the corresponding left and right eigenvectors.听In addition, if the matrices are large and sparse then the computation of the rightmost eigenvalue听can for many classes of nonlinear eigenvalue problems be performed in an efficient way by iterative听algorithms which only rely on matrix vector multiplication and on solving systems of linear equations,听where the structure of the matrices (sparse plus rank one updates) can be exploited.听

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