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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski TakeHome Test 1 Solutions 1 Problem T1.1 Find all values of µ ≤ R for which the function V : R 2 ∞≈ R , defined by ¯ x 1 V ¯ = max { ¯ x 1 , ¯ x 2   x 2 } is monotonically nonincreasing along solutions of the ODE x ˙ 1 ( t ) = µx 1 ( t ) + sin( x 2 ( t )) , x ˙ 2 ( t ) = µx 2 ( t ) − sin( x 1 ( t )) . Answer: µ ∀ − 1. Proof For µ ∀ − 1, x 1 = we have ∈ 1 d 2 2 x 1 = µx 1 + x 1 sin( x 2 ) 2 dt < ( x 1 x 2 ) , − x 1   −  and hence x 1 is strictly monotonically decreasing when x = and x 1 x 2 . Similarly,   ∈   √   x 2 is strictly monotonically decreasing when x = and x 2 x 1 . Hence, when µ ∀ − 1,   ∈   √   V ( x ) is strictly monotonically decreasing along nonequilibrium trajectories of the system. For µ > 1, x 1 (0) = r , x 2 (0) = r , where r > is suﬃciently small we have x ˙ 1 (0) = µr − sin( r ) > , hence V ( x ( t )) √ x 1 ( t ) > r = V ( x (0)) when t > is small enough, which proves that V is not monotonically decreasing. 1 Version of October 20, 2003 2 Problem T1.2 Find all values of r ≤ R for which differential inclusion of the form x ˙( t ) ≤ ( x ( t )) , x (0) = ¯ x , where : R 2 is defined by ∞≈ 2 R 2 (¯ x/  x  ) } for ¯ ∈ x ) = { f (¯ ¯ x = , (0) = { f ( y ) : y = [ y 1 ; y 2 ] ≤ R 2 , y 1 + y 2 ∀ r } ,     has a solution x : [0 , → ) ∞≈ R 2 for every continuous function f : R 2 ∞≈ R 2 and for every initial condition ¯ Answer: r √ 2. x ≤ R 2 ....
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This note was uploaded on 11/07/2011 for the course AERO 16.36 taught by Professor Alexandremegretski during the Spring '09 term at MIT.
 Spring '09
 AlexandreMegretski
 Dynamics

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