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ps6sol_6243_2003

# ps6sol_6243_2003 - Massachusetts Institute of Technology...

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Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 6 Solutions 1 Problem 6.1 For the following statement, verify whether it is true or false (give a proof if true, give a counterexample if false): Assume that (a) n, m are positive integers; (b) f : R n × R m and g : R n × R m are continuously differ- R n R m entiable functions; (c) the ODE y ˙( t ) = g x, y ( t )) has a globally asymptotically stable equilibrium for every x ¯ R n ; (d) functions x 0 : [0 , 1] R n and y 0 : [0 , 1] R m are continuously differentiable and satisfy x ˙ 0 ( t ) = f ( x 0 ( t ) , y 0 ( t )) , g ( x 0 ( t ) , y 0 ( t )) = 0 ± t [0 , 1] . Then there exists 0 > 0 such that for every (0 , 0 ) the differential equation x ˙( t ) = f ( x ( t ) , y ( t )) , y ˙( t ) = 1 g ( x ( t ) , y ( t )) , has a solution x : [0 , 1] R n , y : [0 , 1] R m such that x e (0) = x 0 (0) , y (0) = y 0 (0) , and x ( t ) converges to x 0 ( t ) as 0 for all t [0 , 1] . 1

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ps6sol_6243_2003 - Massachusetts Institute of Technology...

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