The Bohl spectrum for nonautonomous differential equations

We develop the Bohl spectrum for nonautonomous linear differential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in general. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this not evident from the Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum

Some relations between Bohl Exponents and the Exponential Dichotomy spectrum

We study a liaison between the Bohl's exponents and the exponential dichotomy spectrum of a non autonomous linear system of difference equations on the whole line $\mathbb{Z}$. More specifically, We prove that for any initial condition in an invariant vector bundle, associated to its exponential dichotomy spectrum, its Bohl's exponents are contained in an spectral interval.Comment: 9 page

Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

This work proposes a detectability condition for linear time-varying systems based on the exponential dichotomy spectrum. The condition guarantees the existence of an observer, whose gain is determined only by the unstable modes of the system. This allows for an observer design with low computational complexity compared to classical estimation approaches. An extension of this observer design to a class of nonlinear systems is proposed and local convergence of the corresponding estimation error dynamics is proven. Numerical results show the efficacy of the proposed observer design technique

On Lyapunov and Upper Bohl Exponents of Diagonal Discrete Linear Time-Varying Systems

In this paper the necessary and sufficient conditions for two given functions to be the Lyapunov and the upper Bohl exponents of a certain discrete linear system with diagonal coefficients are presented. The obtained conditions are described in terms of easily verifiable properties

On the assignability of regularity coefficients and central exponents of discrete linear time-varying systems

In this paper we investigate the problem of assignability of the so-called regularity coefficients and central exponents of discrete linear time-varying systems. The main result presents a possibility of assignability of Lyapunov, Perron, Grobman regularity coefficients and central exponents by a linear time-varying feedback under the assumptions of uniform complete controllability

On the Sequences Realizing Perron and Lyapunov Exponents of Discrete Linear Time-Varying Systems

We investigate properties of partial exponents (in particular, the Lyapunov and Perron exponents) of discrete time-varying linear systems. In the set of all increasing sequences of natural numbers, we define an equivalence relation with the property that sequences in the same equivalence class have the same partial exponent. We also define certain subclass of all increasing sequences of natural numbers, including all arithmetic sequences, such that all partial exponents are achievable on a sequence from this class. Finally, we show that the Perron and Lyapunov exponents may be approximated by partial exponents achievable on sequences in certain sense similar to geometric sequences

Spectra and leading directions for differential-algebraic equations

The state of the art in the spectral theory of linear time-varying differential-algebraic equations (DAEs) is surveyed. To characterize the asymptotic behavior and the growth rate of solutions, basic spectral notions such as Lyapunov- and Bohl exponents, and Sacker-Sell spectra are discussed. For DAEs in strangeness-free form, the results extend those for ordinary differential equations, but only under additional conditions. This has consequences concerning the boundedness of solutions of inhomogeneous equations. Also, linear subspaces of leading directions are characterized, which are associated with spectral intervals and which generalize eigenvectors and invariant subspaces as they are used in the linear time-invariant setting

On the Error in the Product QR Decomposition

This is the published version, also available here: http://dx.doi.org/10.1137/090761562.We develop both a normwise and a componentwise error analysis for the QR factorization of long products of invertible matrices. We obtain global error bounds for both the orthogonal and upper triangular factors that depend on uniform bounds on the size of the local error, the local degree of nonnormality, and integral separation, a natural condition related to gaps between eigenvalues but for products of matrices. We illustrate our analytical results with numerical results that show the dependence on the degree of nonnormality and the strength of integral separation

Estimating the Relationship between Age Structure and GDP in the OECD Using Panel Cointegration Methods

Economic theory suggests that variations in countries’ age structure should affect the economy on an aggregate level. This paper investigates the relationship between age structure and GDP in 20 OECD countries using annual data from 1970 to 1999. Using new methodology, the relationship between the variables can be formulated in levels despite the presence of unit roots in the time series. Applying two panel cointegration tests proposed by Pedroni (1995, 1997a, 1999), support is found for a long run relationship between GDP and the number of people in five different age groups. Coefficient estimates from panel regressions support effects in line with the life cycle hypothesis and human capital theory; children and retirees are found to have a negative or relatively smaller positive effect on GDP than productive age groups.Age structure; GDP; Panel cointegration