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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 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 ) = x 1 ( t ) + sin( x 2 ( t )) , x 2 ( t ) = x 2 ( t ) sin( x 1 ( t )) . Answer: 1. Proof For 1, x 1 = we have 1 d 2 2 x 1 = x 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 1,       V ( x ) is strictly monotonically decreasing along nonequilibrium trajectories of the system. For > 1, x 1 (0) = r , x 2 (0) = r , where r > is suciently small we have x 1 (0) = r 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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 Spring '09
 AlexandreMegretski
 Dynamics

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