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ABSTRACT a General Linear Systems Theory on Time Scales ABSTRACT A General Linear Systems Theory on Time Scales: Transforms, Stability, and Control Billy Joe Jackson, Ph.D. Advisor: John M. Davis, Ph.D. In this work, we examine linear systems theory in the arbitrary time scale set- ting by considering Laplace transforms, stability, controllability, observability, and realizability. In particular, we revisit the de¯nition of the Laplace transform given by Bohner and Peterson in [10]. We provide su±cient conditions for a given function to be transformable, as well as an inversion formula for the transform. Su±cient 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 sys- tems and before discussing linear time varying systems. The focus is on fundamental notions of linear system control such as controllability, observability, and realizabil- ity. Su±cient 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 suf- ¯cient in the time invariant case. Finally, several notions of stability are discussed, and linear state feedback is explored. A General Linear Systems Theory on Time Scales: Transforms, Stability, and Control by Billy Joe Jackson, B.S., M.A. A Dissertation Approved by the Department of Mathematics Lance L. Littlejohn, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy Approved by the Dissertation Committee John M. Davis, Ph.D., Chairperson Johnny Henderson, Ph.D. Robert J. Marks II, Ph.D. Frank H. Mathis, Ph.D. Qin Sheng, Ph.D. Accepted by the Graduate School August 2007 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on ¯le in the Graduate School. Copyright °c 2007 by Billy Joe Jackson All rights reserved TABLE OF CONTENTS LIST OF FIGURES v LIST OF TABLES vi ACKNOWLEDGMENTS vii 1 A Review of the Time Scales Calculus 1 1.1 Di®erential Calculus . 1 1.2 Integral Calculus . 9 1.3 The Time Scale Exponential Function . 13 1.4 Other Time Scale Functions . 22 2 Laplace Transform 24 2.1 An Overview of the Cases T = R and T = Z . 24 2.2 Introduction to the Arbitrary Time Scale Setting . 28 2.2.1 Properties of the Transform . 38 2.2.2 Inversion Formula . 41 2.2.3 Uniqueness of the Inverse . 49 2.2.4 Frequency Shifting . 51 2.3 Convolution . 53 2.4 The Dirac Delta Functional . 59 2.5 Applications to Green's Function Analysis . 63 3 Linear Systems Theory on Time Scales 65 3.1 Controllability . 65 3.1.1 Time Varying Case . 65 3.1.2 Time Invariant Case . 71 iii 3.2 Observability . 80 3.2.1 Time Varying Case . 80 3.2.2 Time Invariant Case . 83 3.3 Realizability . 87 3.3.1 Time Varying Case . 88 3.3.2 Time Invariant Case . 93 3.4 Stability . 99 3.4.1 Exponential Stability in the Time Invariant Case . 99 3.4.2 BIBO Stability in the Time Varying Case . 104 3.4.3 BIBO Stability in the Time Invariant Case . 108 3.5 Linear Feedback . 113 4 Conclusions and Future Directions 126 BIBLIOGRAPHY 128 iv LIST OF FIGURES 1.1 The Hilger Complex Plane . 4 2.1 Abscissa of Convergence . 32 2.2 Time Varying Hilger Circles . 33 2.3 Commutative Diagram Between Function Spaces . 42 2.4 Commutative diagram in the dual spaces. 61 v LIST OF TABLES 2.1 Laplace Transforms . 39 2.2 Convolutions . 58 vi ACKNOWLEDGMENTS My sincere gratitude goes to John Davis for his time, consideration, and e®ort that he has put into helping me produce this dissertation. He has been a great mentor and I shall always treasure the last three years that we have spent together as advisor and student. Second, I would also like to thank Ian Gravagne and Bob Marks for their contributions to this work. Their valuable input and insight have made this work so much more than it would have been without their ideas. vii CHAPTER ONE A Review of the Time Scales Calculus 1.1 Di®erential Calculus The time scales calculus was ¯rst introduced in the Ph.D. thesis of Stefan Hilger in 1988 (see [32] and [31]). He begins by de¯ning a time scale to be an arbitrary closed subset of the reals, where R is given the standard topology. There is no reason to assume that a time scale be unbounded from above, but we shall make this blanket assumption since all of the results in the following chapters deal with time scales of this type. Let T be a time scale. De¯ne the forward jump operator σ : T ! R by σ(t) := inffs 2 T : s > tg; and the backward jump operator ½ : T ! R by ½(t) := supfs 2 T : s < tg: Here, inf ; = sup T and sup ; = inf T. We de¯ne the graininess function as the distance function between successive points in the time scale, and we denote it by ¹ : T ! R, where ¹(t) = σ(t) ¡ t: The forward derivative or delta derivative of f : T ! R (provided it exists) is then de¯ned as the number f ¢(t) with the property that given any ² > 0, there exists a neighborhood U of t (i.e., U = (t ¡ ±; t + ±) \ T for some ± > 0) such that ¯ ¯ ¯[f(σ(t)) ¡ f(s)] ¡ f ¢(t)[σ(t) ¡ s]¯ · ²jσ(s) ¡ tj for all s 2 U: It is worth noting that this de¯nition is equivalent to the limit f(σ(t)) ¡ f(s) f(σ(t)) ¡ f(t) f ¢(t) = lim = ; s!t σ(t) ¡ s ¹(t) 1 2 as long as the di®erence quotient is interpreted in the limit sense if σ(t) ! t. Points t 2 T with ¹(t) = 0 are termed right-dense, while points with ¹(t) > 0 are called right-scattered. This framework includes two very important cases: namely T = R and T = Z. It is easy to verify that in these two cases, we have σ(t) = t and σ(t) = t + 1, respectively. In each case, the corresponding derivatives are f ¢(t) = f 0(t) and f ¢(t) = ¢f(t), where ¢f(t) := f(t + 1) ¡ f(t) denotes the usual forward di®erence operator. Thus, in this sense, any result concerning the time scales calculus will unify the two classically studied cases. However, there are many other time scales that can now be studied as well. For example, any hybrid set can now also be examined. That is, any set containing points some of which are right-dense with others right-scattered easily ¯t within the framework. Thus, Hilger's calculus will extend the results beyond the cases T = R and T = Z. The time scale [1 Pa;b := [k(a + b); k(a + b) + a] k=0 is an example of a hybrid set since 8 > [1 > <>t; if t 2 [k(a + b); k(a + b) + a); σ(t) = k=0 > [1 > :>t + b; if t 2 fk(a + b) + ag; k=0 and therefore 8 > [1 > <>0; if t 2 [k(a + b); k(a + b) + a); ¹(t) = k=0 > [1 > :>b; if t 2 fk(a + b) + ag: k=0 Besides hybrid sets, Hilger's framework also handles sets which are nonuniformly spaced unlike T = Z. Indeed, one of the most important examples of a time scale besides R and Z is the set T = qZ, which is de¯ned for q > 0 as ( ) T = qZ := qn : n 2 Z [ f0g: 3 This set is commonly referred to as the quantum time scale, and it has a quantum calculus that has been studied in the literature (see [7], [6], and [14]). Notice that this time scale does indeed exhibit hybrid features since t = 0 is right dense and every other point is right scattered and nonuniformly spaced. In what follows, we shall give the most pertinent results from the time scales calculus that will be needed in later chapters. We will focus on the matrix case, as the scalar case will of course be a special case of the matrix case. A complete treatment of the calculus can be found in Bohner and Peterson in [8] and [9]. We begin by developing the geometry of the Hilger complex plane. For ¹(t) > 0, the Hilger complex numbers, the Hilger real axis, the Hilger alternating axis, and the Hilger imaginary circle (or simply the Hilger circle) are all respectively de¯ned as ( ) 1 C := z 2 C : z 6= ¡ ¹ ¹(t) ( ) 1 R := z 2 C : z 2 R and z > ¡ ; ¹ ¹ ¹(t) ( ) 1 A := z 2 C : z 2 R and z < ¡ ; ¹ ¹ ¹(t) ( ¯ ¯ ) ¯ 1 ¯ 1 H := z 2 C : ¯z + ¯ = ; ¹ ¹ ¯ ¹(t)¯ ¹(t) and for ¹(t) = 0, de¯ne C0 := C; R0 := R; H0 := iR; and A0 := ;. For any z 2 C¹, the Hilger real part of z is given by jz¹(t) + 1j ¡ 1 Re (z) := ; ¹ ¹(t) while the Hilger imaginary part of z is de¯ned as Arg(z¹(t) + 1) Im (z) := ; ¹ ¹(t) where Arg(z) is the principal argument of z. For ¹(t) = 0, de¯ne Re¹(z) = Re(z) ¼ and Im¹(z) = Im(z). The Hilger purely imaginary number º¶! is de¯ned for ¡ ¹(t) · 4 Figure 1.1: The Hilger Complex Plane.
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