rec6 - || . Exercise Consider the following two systems of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Sam Burden Linear Systems ( EE 221 ) Recitation #6 — Oct 1, 2010 1 Inner Product Spaces Consider ( H,F, , ·i ). Do not assume H is complete, i.e. a Hilbert Space. Exercise The inner product , ·i : H × H F is continuous. (a) Define continuity in this context. (b) Show that |h x i ,y j i - h x,y i| ≤ | x i - x || y j - y | + | x i - x || y | + | x || y j - y | . (c) Show that , ·i is continuous. Self-Adjoint Maps Let A : U V continuous and linear with U , V Hilbert spaces. Exercise A * : V U is linear and continuous with induced norm | A * | = | A | . Definition A : U U is self-adjoint if A = A * , i.e. x,y U : h x,Ay i = h Ax,y i . Exercise Let A : U U be self-adjoint. (a) All eigenvalues of A are real. (b) If ( v j j ), ( v k k ) are eigenpairs and λ j 6 = λ k , h v j ,v k i = 0. Lipschitz Continuity Fact | det x f ( x,t ) | ≤ K ( t ), piecewise continuous K ( t ): x,y R n : t 0 : || f ( x ) - f ( y ) || ≤ K ( t ) || x - y
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: || . Exercise Consider the following two systems of dierential equations: x 1 =-x 1 + e t cos( x 1-x 2 ) x 2 =-x 2 + 15 sin( x 1-x 2 ) , x 1 =-x 1 + x 1 x 2 x 2 =-x 2 . (a) Do they satisfy a global Lipschitz condition? (b) For the second system, are solutions uniquely dened for all possible initial conditions? State Transition Matrix Exercise Let ( t, ) be the state transition matrix of x = A ( t ) x . (a) For nonsingular M ( t ) R n n , determine an expression for d dt M-1 ( t ). (b) Using this result, nd an expression for d d ( t, )....
View Full Document

Ask a homework question - tutors are online