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t2sol_6243_2003

# t2sol_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 Take-Home Test 2 Solutions 1 Problem T2.1 System of ODE equations x ˙( t ) = Ax ( t ) + ( Cx ( t ) + cos ( t )) , (1.1) where A, B, C are constant matrices such that CB = 0 , and ψ : R k is ∈� R q continuously differentiable, is known to have a locally asymptotically stable non-equilibrium periodic solution x = x ( t ) . What can be said about trace( A ) ? In other words, find the set of all real numbers such that = trace( A ) for some A, B, C, ψ such that (1.1) has a locally asymptotically stable non-equilibrium periodic solution x = x ( t ) . Answer: trace( A ) < 0. Let x 0 ( t ) be the periodic solution. Linearization of (1.1) around x 0 ( · ) yields α ˙ ( t ) = ( t ) + Bh ( t ) ( t ) , where h ( t ) is the Jacobian of ψ at x 0 ( t ), and x ( t ) = x 0 ( t ) + α ( t ) + o ( α ( t ) ) . | | Partial information about local stability of x 0 ( · ) is given by the evolution matrix M ( T ), where T > 0 is the period of x 0 ( · ): if the periodic solution is asymptotically stable then all eigenvalues of M ( T ) have absolute value not larger than one. Here M ˙ ( t ) = ( A + Bh ( t ) C ) M ( t ) , M (0) = I, 1 Version of November 25, 2003

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2 and hence ⎛� T det M ( T ) = exp trace( A + Bh ( t ) C ) dt . 0 Since trace( A + Bh ( t ) C ) = trace( A + CBh ( t )) = trace( A ) , det( M ( T )) > 1 whenever trace( A ) > 0. Hence trace( A ) 0 is a necessary condition for local asymptotic stability of x 0 ( · ). Since system (1.1) with k = q = 1, ψ ( y ) y , A = a 0 1 , B = , C = 0 1 0 a 0 has periodic stable steady state solution (1 + a 2 ) 1 cos( t ) + a (1 + a 2 ) 1 sin( t )) x 0 ( t ) = 0 for all a > 0, the trace of A can take every negative value. Thus, to complete the solution, one has to figure out whether trace of A can take the zero value. It appears that the volume contraction techniques are better suited for solving the question completely. Indeed, consider the autonomous ODE z ˙ 1 ( t ) = z 2 ( t ) , z ˙ 2 ( t ) = z 1 ( t ) , (1.2) z 1 ( t ) z ˙ 3 ( t ) = Az 3 ( t ) + Cz 3 ( t ) + z 1 ( t ) 2 + z 2 ( t ) 2 , 2 defined for z 1 + z 2 2 = 0. If (1.1)
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t2sol_6243_2003 - Massachusetts Institute of Technology...

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