lec8_6243_2003

lec8_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 Lecture 8: Local Behavior at Eqilibria 1 This lecture presents results which describe local behavior of autonomous systems in terms of Taylor series expansions of system equations in a neigborhood of an equilibrium. 8.1 First order conditions This section describes the relation between eigenvalues of a Jacobian a x 0 ) and behavior of ODE x ˙( t ) = a ( x ( t )) (8.1) or a difference equation x ( t + 1) = a ( x ( t )) (8.2) in a neigborhood of equilibrium ¯ x 0 . In the statements below, it is assumed that a : X R n is a continuous function defined on an open subset X ± R n . It is further assumed that ¯ x 0 X , and there exists an n -by- n matrix A such that | a x 0 ) | x 0 + λ ) a 0 as | λ | 0 . (8.3) | λ | If derivatives da k /dx i of each component a k of a with respect to each cpomponent x i of x exist at ¯ x 0 , A is the matrix with coefficients da k /dx i , i.e. the Jacobian of the system. However, differentiability at a single point ¯ x 0 does not guarantee that (8.3) holds. On the other hand, (8.3) follows from continuous differentiability of a in a neigborhood of ¯ x 0 . 1 Version of October 3, 2003
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± ²³ 2 Example 8.1 Function a : R 2 R 2 , defined by ¯ ¯ 2 ¯ 2 2 x 2 ) 2 ± ¯ x 1 x 1 x 2 x 1 ¯ 2 x 1 ² a = ¯ 2 ¯ 2 x 2 ¯ ¯ x 2 x 1 x 2 + 2 ¯ 2 ) 2 x 2 x 1 for x x 1 and ¯ ¯ = 0, and by a (0) = 0, is differentiable with respect to ¯ x 2 at every point ¯ x R 2 , and its Jacobian a (0) = A equals minus identity matrix. However, condition (8.3) is not satisfied (note that a x = 0). x ) is not continuous at ¯ 8.1.1 The continuous time case Let us call an equilibrium ¯ x 0 of (8.1) exponentially stable if there exist positive real num- bers σ, r, C such that every solution x : [0 , T ] X with | x (0) ¯ x 0 | < σ satisfies x 0 | ∀ Ce rt | x (0) ¯ | x ( t ) ¯ x 0 | ± t 0 . The following theorem can be attributed directly to Lyapunov. Theorem 8.1 Assume that a x 0 ) = 0 and condition (8.3) is satisfied. Then (a) if A = a x 0 ) is a Hurwitz matrix (i.e. if all eigenvalues of A have negative real part) then ¯ x 0 is a (locally) exponentially stable equilibrium of (8.1); x 0 ) has an eigenvalue with a non-negative real part then ¯ (b) if A = a x 0 is not an exponentially stable equilibrium of (8.1); x 0 ) has an eigenvalue
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lec8_6243_2003 - Massachusetts Institute of Technology...

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