A note on the perturbation of positive matrices by normal and unitary matrices

AbstractIn a recent paper, Neumann and Sze considered for an n×n nonnegative matrix A, the minimization and maximization of ρ(A+S), the spectral radius of (A+S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n×n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A+N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A+N) as N ranges over all such matrices. We show that the extremal values always occur at an n×n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A+X) over all n×n real matrices X of Frobenius norm n was sought

Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems

We study a growth maximization problem for a continuous time positive linear system with switches. This is motivated by a problem of mathematical biology (modeling growth-fragmentation processes and the PMCA protocol). We show that the growth rate is determined by the non-linear eigenvalue of a max-plus analogue of the Ruelle-Perron-Frobenius operator, or equivalently, by the ergodic constant of a Hamilton-Jacobi (HJ) partial differential equation, the solutions or subsolutions of which yield Barabanov and extremal norms, respectively. We exploit contraction properties of order preserving flows, with respect to Hilbert's projective metric, to show that the non-linear eigenvector of the operator, or the "weak KAM" solution of the HJ equation, does exist. Low dimensional examples are presented, showing that the optimal control can lead to a limit cycle.Comment: 8 page

Stability radii of positive linear Volterra–Stieltjes equations

AbstractWe study stability radii of linear Volterra–Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra–Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new

A note on the eigenvectors of perturbed matrices with applications to linear positive systems

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms — acting much like Lyapunov functions — for linear positive systems, which mayhelp estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples