The notions of transfer matrix, transfer equivalence, and input-output equivalence for linear control systems on time scales are introduced. These concepts generalize the corresponding continuous- and discrete-time versions. Necessary and sufficient conditions for transfer and input-output equivalence are presented. As the main tool, an extension of the Laplace transform for functions defined on a time scale is used.

We give a comprehensive treatment of Sturm–Liouville operators with measure-valued coefficients including, a full discussion of self-adjoint extensions and boundary conditions, resolvents, and Weyl–Titchmarsh theory. We avoid previous technical restrictions and, at the same time, extend all results to a larger class of operators. Our operators include classical Sturm–Liouville operators, Lax operators arising in the treatment of the Camassa–Holm equation, Jacobi operators, and Sturm–Liouville operators on time scales as special cases.

We prove a necessary and sucient condition for the exponential stability of time-invariant linear systems on time scales in terms of the eigenvalues of the system matrix. In particular, this unifies the corresponding characterizations for finite-dimensional differential and difference equations. To this end we use a representation formula for the transition matrix of Jordan reducible systems in the regressive case. Also we give conditions under which the obtained characterizations can be exactly calculated and explicitly calculate the region of stability for several examples.

We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection we generalize several results for Sturm–Liouville equations on time scales which have been obtained by various authors in the past.

Abstract. We show that the Hilger derivative on time scales is a special case of the Radon–Nikodym derivative with respect to the natural measure associated with every time scale. Moreover, we show that the concept of delta absolute continuity agrees with the one from measure theory in this context. 1.

It has been known for some time that proportional output feedback will stabilize certain classes of linear time-invariant systems under an adaptation mechanism that drives the feedback gain su#ciently high. More recently, it was demonstrated that discrete implementations of the high-gain adaptive controller also require adaptation of the sampling rate. In this paper, we use recent advances in the mathematical field of dynamic equations on time scales to unify the discrete and continuous versions of the high-gain adaptive controller. A novel proof method is presented based on time scales, as is a brief tutorial on the subject of time scales.

We prove the Grüss inequality on time scales and thus unify corresponding continuous and discrete versions from the literature. We also apply our results to the quantum calculus case. AMS Subject Classification: 39A10; 39A12; 39A13.

Cataloguing-in-Publication entry

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Abstract. In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system’s poles in the complex plane.