The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. In this paper, we prove the second Euler-Lagrange necessary optimality condition for optimal trajectories of variational problems on time scales. As an example of application of the main result, we give an alternative and simpler proof to the Noether theorem on time scales recently obtained in [J. Math. Anal. Appl. 342 (2008), no. 2, 1220–1226].

Nonlinear partially defined systems on an arbitrary unbounded time scale are studied. They include continuous-time and discrete-time systems. The main problem is to find necessary and sufficient conditions for an abstract input/output map to have a realization as a nonlinear system of a specific class on the time scale. The obtained results extend criteria of realizability of continuous-time polynomial systems. A simple construction of a realization is provided.

Positive linear systems on arbitrary time scales are studied. The theory developed in the paper unifies and extends concepts and results known for continuous-time and discrete-time systems. A necessary and sufficient condition for a linear system on a time scale to be positive is presented. Properties of positive reachable sets are investigated and characterizations of various controllability properties are presented. A modified Gram matrix of the system is used to state necessary and sufficient condition of positive reachability of a positive system on an arbitrary time scale.