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Mathematics for Control Theory Mathematics for Control Theory Elements of Analysis p.1 Metric Spaces Contraction Mapping Principle Hanz Richter Mechanical Engineering Department Cleveland State University Reading materials We will use: ■ Anthony N. Michel and Charles J. Herget (2007) [1981], Algebra and Analysis for Engineers and Scientists, Birkhäuser, e-ISBN-13: 978-0-8176-4707-0 (chapter 5, skip sections indicated by the instructor) ■ Kenneth R. Davidson and Allan P. Donsig [2002], Real Analysis with Real Applications, Prentice-Hall, ISBN 0-13-041647-9 (use chapter 4 as a reference). 2 / 30 Introduction Metric spaces are the most general setting in which we can study important aspects of sequences and functions such as convergence and continuity. Essential concepts pertaining to the base sets of metric spaces, such as openness and compactness and their relation to functions defined on them will be covered in this section. Metric spaces are sets where a distance function (the metric) is defined for any pair of its elements. Only a few requirements are made for the metric; otherwise, the definition is very general. 3 / 30 Definition of metric space Let X be any nonempty set and let ρ be a real-valued function defined on X X satisfying: × 1. ρ(x, y) 0 x, y X and ρ(x, y)=0 x = y. ≥ ∀ ∈ ⇐⇒ 2. ρ(x, y) = ρ(y,x) x, y X ∀ ∈ 3. ρ(x, y) ρ(x,z) + ρ(z, y) x, y X ≤ ∀ ∈ Metric spaces defined as above will be denoted X; ρ . A given X admits { } infinitely many metrics. Examples: ρ(x, y) = x y is a metric when X = R. Also, | − | ρ(z1,z2) = (z1 z2)(z1 z2) is a metric when X = C. − − q 4 / 30 Discrete metric, diameter The following function is a metric (show) which works for any X: 0, x = y ρ(x, y) = 1, x = y 6 We say that a metric space X; ρ is bounded if M > 0 s.t. ρ(x, y) M { } ∃ ≤ x, y X (the metric is bounded on X). ∀ ∈ The diameter of a subset Y X in a metric space X; ρ is defined by ∈ { } δ(Y ) = sup ρ(x, y) : x, y Y { ∈ } We solve Ex.5.1.20 in Michel and Herget: an inclusion relationship among subsets of a metric space implies an order relationship in the corresponding diameters. Is the opposite true? 5 / 30 Important metric spaces We can make a metric space with complex n-tuples (Cn, with Rn a particular case) and a very general metric. Let x =(ξ1,ξ2,...ξn) and y =(η1, η2,...ηn). Define the p-metric as 1 n p p ρp(x, y) = ξi ηi | − | "i=1 # X for 1 p < . ≤ ∞ This metric satisfies Minkowski’s inequalities: ρp(x, y) ρp(x, 0) + ρp(y, 0) ≤ ρp(x, y) ρp(x, 0) + ρp(y, 0) − ≤ When p =2, we recover the usual Euclidean distance. Matlab: ρp(x, 0) = norm(x,p) . 6 / 30 p-metrics... When x and y are infinite sequences rather than vectors in Cn, we can use the above metric for n = provided the sequences converge (see Ex. 5.3.5 in ∞ Michel and Herget): ∞ ξ p < | |i ∞ i=1 X ∞ η p < | |i ∞ i=1 X When p =1, we obtain an absolute value measure of distance. At the other end of the range of p, we define for x, y Cn: ∈ ρ∞(x, y) = max ξi ηi {| − |} Show that this is indeed a metric space. 7 / 30 Distance between continuous functions Another important metric space can be formed with all real-valued continuous functions on an interval [a, b]. This set is denoted as [a, b]. The following C metric works: b p p ρp(x, y) = x(t) y(t) dt | − | Za for all x, y [a, b]. ∈C Homework: (0.5pt) Solve 5.3.18 in Michel and Herget. 8 / 30 Product Metric Spaces How would we capture objects such as vectors whose entries themselves are elements of metric spaces? What are possible metrics? n Let X1; ρ1 , X2; ρ2 .... Xn; ρn be n metric spaces. Let X = Xi. { } { } { } i=1 Consider the following metrics: Q n ρ(x, y) = ρi(xi, yi) i=1 X 1 n 2 2 ρ(x, y) = (ρi(xi, yi)) i=1 ! X for any x =(x1,x2...xn) and y =(y1, y2,...yn) in X. Th. 5.3.21 shows that both metrics can be used to form metric spaces on X. 9 / 30 Open and Closed Sets This topic is of the highest importance and should remain with you forever. We will introduce the topological definition later in the course. Let X; ρ be any metric space. { } Definition: Let x0 X and let r be a positive real. The open ball S(x0; r) ∈ around x0 is the set: S(x0; r) = x X : ρ(x,x0) < r { ∈ } An equivalent terminology is to say that S is a ball of radius r around x0. Definition: A subset Y of X is open if for every x X there exists some open ∈ ball S(x, r) such that S(x, r) Y . ⊂ X Y Z 10 / 30 Closed Sets Again let Y X. ⊂ Definition: A point x X is called an adherent point of Y if every open ball ∈ with center x contains at least one point of Y . The set of all adherent points of Y is called the closure of Y , denoted Y¯ . Definition: Y is a closed subset of X is Y¯ = Y . Are the following implications true?: A is not open A is closed ⇒ A is not closed A is open ′⇒ A is open A is closed. ⇒ Homework: (0.5pt) Solve 5.4.26 in Michel and Herget. 11 / 30 Important results Let X; ρ be a metric space. { } Theorem (5.4.15 in Michel and Herget): 1. X and ∅ are open 2. Let Yα be an arbitrary family of open subsets of X where α belongs to { } some set A. Then α∈A Sα is open. 3. The intersection ofS a finite number of open sets of X is open. In class, we give an example of 2. and a counterexample for 3. when the intersection involves a non-finite set of subsets of X. 12 / 30 Important results... Theorem (5.4.18 - ii in Michel-Herget): If Y X is open, there is a family of ⊂ open balls Sα α∈A such that Y = Sα. { } α∈A S Proof (by double inclusion) : Let x Y . Since Y is open, x is the center of ∈ some open ball Sx contained in Y . Then x Sx. All Sx form the required ∈ family. Now assume x Sα. Then x Sα for some α A, and Sα Y ∈ α∈A ∈ S ∈ ∈ because Sα is an open subset of Y (Th. 5.4.18 -i). Therefore x Y . S ∈ Examine Theorem 5.4.27 independently. These results will be recalled in upcoming material or exercises. Definition: Let x0 X and r > 0. The closed sphere centered at x0 with ∈ radius r is defined as: K(x0; r) = x X : ρ(x0,x) r { ∈ ≤ } It is actually closed (Th. 5.4.27 -ii). 13 / 30 Sequences on Metric Spaces The following material is relative to a metric space X; ρ . We denote the + { } positive integers as Z . + Definition: A sequence xn in a set Y X is a function f : Z Y , so { } ⊂ → that f(n) = xn. Definition (convergence): Sequence xn converges to x X if ǫ > 0, + { } ∈ ∀ N Z s.t. ρ(xn,x) <ǫ n N. ∃ ∈ ∀ ≥ With convergence, we write: lim xn = x n→∞ Example: an : n =0, 1, 2.. converges to zero for a < 1 (show using above { } | | definition). 14 / 30 Cauchy Sequences We can study convergence without involving the value of the limit. This is important because there are sequences whose limit is outside X (these appear in incomplete metric spaces). Definition: A sequence xn in a metric space X; ρ is Cauchy if + { } { } ǫ > 0, N Z s.t. ρ(xn,xm) ǫ whenever n, m N. ∀ ∃ ∈ ≤ ≥ This definition says that in a Cauchy sequence, any pair of elements coming at or after an index N will be separated by a distance no greater than some ǫ. Theorem (5.5.12): If xn converges then it is Cauchy. { } Theorem (5.5.14) If a Cauchy sequence xn contains a convergent { } subsequence, then xn converges. { } The proof is slightly more elaborate than previous ones in the course. We develop it in class. 15 / 30 Complete Metric Spaces Definition: If every Cauchy sequence in a metric space X; ρ converges to { } some element of X, then the metric space is complete. This is an important definition. We will see in the next example that X = [a, b] (with the ρ2 metric) contains function sequences which converge C to discontinuous functions. Oddly enough, completeness is obtained with the ρ∞ metric. This is one reason to seek more inclusive spaces that allow discontinuous functions, crucial in control theory. Relaxing continuity to Lebesgue measurability will allow us to form a complete metric space with ρ2 as metric. We work on various examples at this point. 16 / 30 Compactness We will focus on metric spaces based on Cn (Rn as a particular case). Definition: Let X; ρ be a metric space and let Y X. A collection of { } ⊂ subsets Yα , α A of X is called a covering of Y if Y Yα. A { } ∈ ⊂ α∈A subcollection Yβ , β B, B A such that Y Yβ is called a { } ∈ ⊂ ⊂ β∈B S subcovering of Yα . { } S If Yα and Yβ are open, the covering and the subcovering are open.
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