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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 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 ) = . The linearized system is asymptotically stable if and only if < cos( a ) < . 5. Hence the equilibrium y ( t ) a of the original system is locally exponentially stable if and only if < cos( a ) < . 5. Problem 5.2 In order to solve a quadratic matrix equation X 2 + AX + B = 0 , where A, B are given nby n matrices and X is an nby 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 = . 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 2 exponentially as k when X X is small enough? You are allowed to use the fact that matrix equation...
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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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