Abstract
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.
Highlights
In many applications the variables that appear in a mathematical description take only positive or nonnegative values
We study two controllability properties of positive linear systems: positive accessibility and positive reachability
The characterizations hold on arbitrary time scales, but they have different features
Summary
In many applications the variables that appear in a mathematical description take only positive or nonnegative values. The concepts of standard derivative used in the case of continuous time and forward difference used in the discrete time are unified into one concept of delta derivative This allows to consider delta differential equations on arbitrary time scales. Special attention was paid to linear delta differential equations Another theory unifying discrete and continuous dynamical systems was developed in [19]. Criteria of positive reachability for discrete-time systems and continuous-time systems are essentially different. As the criteria for positive reachability are completely different for continuous-time and discrete-time systems, we have tried to develop methods that would result in the same statements for different time scales. We state reachability criteria for a discrete nonhomogeneous time scale and for the time scale that is a union of disjoint closed intervals The latter differs significantly from the standard continuous-time case.
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