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

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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 , λ j ), ( v k , λ 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 ) || ≤
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Unformatted text preview: || . Exercise Consider the following two systems of differential 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 defined for all possible initial conditions? State Transition Matrix Exercise Let Φ( t,τ ) 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, find an expression for d dτ Φ( t,τ )....
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