| In this work, we examine linear systems theory in the arbitrary time scale setting by considering Laplace transforms, stability, controllability, observability, and realizability. In particular, we revisit the definition of the Laplace transform given by Bohner and Peterson in [10]. We provide sufficient conditions for a given function to be transformable, as well as an inversion formula for the transform. Sufficient conditions for the inverse transform to exist are provided, and uniqueness of this inverse function is discussed. Convolution under the transform is then considered. In particular, we develop an analogue of the Convolution Theorem for arbitrary time scales and discuss the algebraic properties of the convolution. This naturally leads to an algebraic identity for the convolution operator, which is a time scale analogue of the Dirac delta distribution.;Next, we investigate applications of the transform to linear time invariant systems and before discussing linear time varying systems. The focus is on fundamental notions of linear system control such as controllability, observability, and realizability. Sufficient conditions for a system to possess each of these properties are given in the time varying case, while these same criteria often become necessary and sufficient in the time invariant case. Finally, several notions of stability are discussed, and linear state feedback is explored. |
