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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- Fall '10
- Linear Algebra, Continuous function, state transition matrix, Hilbert space, Inner product space, Lipschitz continuity