Ivica Nakić

Wielandt and Ky-Fan theorem for matrix pairs

The generalization of Wielandt and Ky-Fan theorem is given for Hermitian matrix pairs, and some new eigenvalue perturbation estimates are obtained. An application is made on a class of quadratic matrix pencils.

Scale-free uncertainty principles and Wegner estimates for random breather potentials ☆

Abstract We present new scale-free quantitative unique continuation principles for Schrodinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veselic [15] , and Klein [10] . We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schrodinger operator of breather type. It holds for arbitrarily high energies. Schrodinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence.

Lyapunov optimization of a damped system

Our aim is to optimize the damping of a linear vibrating system. The penalty function is the average total energy, which is equal to the trace of the corresponding Lyapunov solution. We prove the existence and the uniqueness of the global minimum, if the damping varies over the set of all possible positive definite matrices. The minimum is shown to be taken on the so-called modal critical damping, thus confirming a long existing conjecture. We also give some preliminary results concerning dampings which depend linearly on the viscosity parameters whereas the damper positions are kept fixed. We produce physical examples on which the minimum is taken on a negative viscosity or which have several local minima. Both phenomena seem to be a consequence of a bad choice of the damper positions.

Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector χ ( − ∞ , E ] ( H L ) χ(−∞, E](HL) of a Schrodinger operator H L HL on a cube of side L ∈ N L∈ℕ, with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors χ ( E − γ , E ] ( H L ) χ(E−γ, E](HL) with small γ γ. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrodinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.

Relative perturbation theory for a class of diagonalizable Hermitian matrix pairs

Abstract Veselic and Slapnicar gave a general perturbation result for the eigenvalues of the Hermitian matrix pair ( H , K ), where K is positive definite. In this paper their result is generalized to a wider class of Hermitian matrix pairs. Especially, estimates for the relative perturbation of eigenvalues of definite pairs are also obtained.

Multiscale Unique Continuation Properties of Eigenfunctions

Quantitative unique continuation principles for multiscale structures are an important ingredient in a number applications, e.g., random Schrodinger operators and control theory.

Optimal Direct Velocity Feedback

We present a novel approach to the problem of Direct Velocity Feedback (DVF) optimization of vibrational structures, which treats simultaneously small as well as large gains. For that purpose, we use two different approaches. The first one is based on the gains optimization using the Lyapunov equation. In the scope of this approach we present a new formula for the optimal gain and we present a relative error for modal approximation. In addition, we present a new formula for the solution of the corresponding Lyapunov equation for the case with multiple undamped eigenfrequencies, which is a generalization of existing formulae. The second approach studies the behavior of the eigenvalues of the corresponding quadratic eigenvalue problem. Since this approach leads to the parametric eigenvalue problem we consider small and large gains separately. For the small gains, which are connected to a modal damping approximation, we present a standard approach based on Gerschgorin discs. For the large gains we present a new approach which allows us to approximate all eigenvalues very accurately and efficiently.