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ps5sol_6243_2003

# ps5sol_6243_2003 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 5 Solutions 1 Problem 5.1 y ( t ) a is an equilibrium solution of the differential equation y (3) ( t ) + ¨ y ( t ) + ˙ y ( t ) + 2 sin( y ( t )) = 2 sin( a ) , where a R and y (3) denotes the third derivative of y . For which values of a R is this equilibrium locally exponentially stable? The linearized equations for small ( t ) = y ( t ) a are given by (3) ( t ) + ¨ ( t ) + ˙ ( t ) + 2 cos( a ) ( t ) = 0 . The linearized system is asymptotically stable if and only if 0 < cos( a ) < 0 . 5. Hence the equilibrium y ( t ) a of the original system is locally exponentially stable if and only if 0 < cos( a ) < 0 . 5. Problem 5.2 In order to solve a quadratic matrix equation X 2 + AX + B = 0 , where A, B are given n -by- n matrices and X is an n -by- n matrix to be found, it is proposed to use an iterative scheme X k +1 = X k 2 + AX k + X k + B. Assume that matrix X satisfies X 2 + AX + B = 0 . What should be required of the eigenvalues of X and A + X in order to guarantee that X k X 1 Version of November 12, 2003

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2 exponentially as k � ≈ when X 0 X is small enough? You are allowed to use the fact that matrix equation ay + yb = 0 , where a, b, y are n -by- n matrices, has a non-zero solution y if and only if det( sI a ) = det( sI + b ) for some s C .
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ps5sol_6243_2003 - Massachusetts Institute of Technology...

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