t1sol_6243_2003

t1sol_6243_2003 - Massachusetts Institute of Technology...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 Take-Home 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 non-increasing 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 non-equilibrium 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 ....
View Full Document

Page1 / 5

t1sol_6243_2003 - Massachusetts Institute of Technology...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online