In this work, we reexamine the time scale Laplace transform as defined by Bohner and Peterson [M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001; M. Bohner, A. Peterson, Laplace transform and Z-transform: Unification and extension, Methods Appl. Anal. 9 (1) (2002) 155-162]. In particular, we give conditions on the class of functions which have a transform, develop an inversion formula for the transform, and further, we provide a convolution for the transform. The notion of convolution leads to considering its algebraic structure-in particular the existence of an identity element-motivating the development of the Dirac delta functional on time scales. Applications and examples of these concepts are given.

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Abstract-For LTI systems on a class of nonuniform discrete domains, we establish a region in the complex plane for which pole placement is a necessary and sufficient condition for exponential stability of solutions of the system. We study the interesting geometry of this region, comparing and contrasting it with the standard geometry of the regions of exponential stability for ODE systems on R and finite difference/recursive equations on Z. This work connects other results in the literature on the topic and explains the connection geometrically using time scales theory.Index Terms-exponential stability, pole placement, time scales I. EXPONENTIAL STABILITY ON R AND ZLet A ∈ R n×n . A basic result concerning the continuous, linear time invariant (LTI) systeṁis that solutions are exponentially stable if and only if spec(A) ⊂ C − . A similar basic result for discrete LTI systemsis that solutions are exponentially stable if and only if spec(A) ⊂ {z ∈ C : |z| < 1}. An equivalent reformulation is that solutions ofare exponentially stable if and only if spec(A) ⊂ {z ∈ C : |1+ z| < 1}. Here, ∆ denotes the forward difference operator. For reasons that will soon become apparent, we will use (I.3) rather than (I.2) as the canonical discrete LTI system throughout this paper. Thus, the regions of exponential stability for (I.1) and (I.3) are quite straightforward. This simple geometry is exploited frequently in pole placement arguments for exponential stability . In this paper, we explore the following question: What is the geometry of the region of exponential stability for an LTI system defined on a nonuniform discrete domain?II. EXPONENTIAL STABILITY OF NONUNIFORM DISCRETE SYSTEMS This question can be efficiently handled using time scales theory [5]; see the Appendix for a brief overview. Let T be a nonuniform, discrete time scale that is unbounded above, and consider the LTI systemor its equivalent recursive formDefinition II.1.[25] For t, t 0 ∈ T and x 0 ∈ R n , the systemis exponentially stable if there exists a constant α > 0 such that for every t 0 ∈ T there exists a K ≥ 1 withwith K being chosen independently of t 0 . Here, Φ A (t, t 0 ) denotes the unique solution to (II.3), also called the transition matrix for (II.3); see the Appendix.The following theorem due to Pötzsche, Siegmund, and Wirth [25] provides a spectral characterization of the region of exponential stability of (II.2) for scalar problems.Theorem II.1.[25] Let T be a time scale which is unbounded above. Fix t 0 ∈ T and let λ ∈ C. Then the scalar equationis exponentially stable if and only if either of the following holds:(C2) For every T ∈ T, there exists a t ∈ T with t > T such that 1 + µ(t)λ = 0, where we use the convention log 0 = −∞ in (C1).Definition II.2.[25] Given a time scale T which is unbounded above, for arbitrary t 0 ∈ T, define the sets S C (T) := {λ ∈ C : (C1) holds}, S R (T) := {λ ∈ R : (C2) holds}.