This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Nonlinear Systems and Control Lecture # 0 Mathematical Preliminaries – p.1/39 Vector Space (or Linear Space): A vector space ( V , F ) consists of a set (of vectors) V , a field (of scalars) F and two operations viz. addition of vectors (+) and multiplication of vectors by scalars ( · ) , which obey the following axioms: 1. Addition is given by + : V × V → V : ( x + y ) → x + y ; It is Associative: ( x + y ) + z = x + ( y + z ) , ∀ x,y,z ∈ V Commutative: x + y = y + x, ∀ x,y ∈ V ∃ ! identity , (called the zero vector), s.t. x + 0 = 0 + x = x, ∀ x ∈ V ∃ ! inverse: ∀ x ∈ V , ∃ !( − x ) ∈ V s.t. x + ( − x ) = 0 ; – p.2/39 2. Multiplication by scalars is given by · : F × V → V ( α, x ) → αx , where ∀ x ∈ V and ∀ α,β ∈ F , ( αβ ) x = α ( βx ) 1 x = x , x = 0 ; 3. Addition and multiplication by scalars are related by distributed laws viz. ∀ x ∈ V , ∀ α,β ∈ F ( α + β ) x = αx + βx ∀ x,y ∈ V , ∀ α ∈ F α ( x + y ) = αx + αy . Subspace: Let W be a subset of V . If W is a vector space itself, with the same vector space operations as V has, then it is a subspace of V . – p.3/39 Inner Product Space: A vector space ( V , F ) is an inner product space if there is a function (· , ·) : V × V mapsto→ F such that for every x,y,z ∈ V and α ∈ F the following hold: ( x + y,z ) = ( x, z ) + ( y,z ) , ( x,y + z ) = ( x, y ) + ( x,z ) ( x,αy ) = α ( x,y ) ∀ α ∈ F bardbl x bardbl 2 = ( x,x ) ≥ ( x,x ) = 0 if and only x = 0 . ( x,y ) = ( y,x ) , where the overbar denotes the complex conjugate operator. The function (· , ·) is called the inner product on V . – p.4/39 The supremum or least upper bound (LUB) of a set S of real numbers is denoted by sup( S ) and is defined to be the smallest real number that is greater than or equal to every number in S . Every nonempty subset of the set of real numbers that is bounded above has a supremum that is also an element of the set of real numbers. sup { x ∈ R : 0 < x < 1 } = sup { x ∈ R : 0 ≤ x ≤ 1 } = 1 . The infimum or greatest lower bound (GLB) of a set S of real numbers is denoted by inf( S ) and is defined to be the biggest real number that is smaller than or equal to every number in S . Any bounded nonempty subset of the real numbers has an infimum in the nonextended real numbers. inf { x ∈ R : 0 < x < 1 } = 0 . – p.5/39 Vector Norms: Let X be a vector space, a realvalued function bardbl · bardbl defined on X is said to be a norm on X if it satisfies the following properties: bardbl x bardbl ≥ (positivity) ; bardbl x bardbl = 0 if and only if x = 0 (positive definiteness) ; bardbl αx bardbl =  α bardbl x bardbl , for any scalar α (homogeneity) ; bardbl x + y bardbl negationslash = bardbl x bardbl + bardbl y bardbl (triangle inequality) for any x ∈ X and y ∈ X ....
View
Full Document
 Spring '08
 CHOI
 Topology, Continuous function, Metric space, Compact space, Banach space

Click to edit the document details