lec11_6243_2003

# lec11_6243_2003 - Massachusetts Institute of Technology...

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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 Lecture 11: Volume Evolution And System Analysis 1 Lyapunov analysis, which uses monotonicity of a given function of system state along trajectories of a given dynamical system, is a major tool of nonlinear system analysis. It is possible, however, to use monotonicity of volumes of subsets of the state space to predict certain properties of system behavior. This lecture gives an introduction to such methods. 11.1 Formulae for volume evolution This section presents the standard formulae for evolution of volumes. 11.1.1 Weighted volume Let U be an open subset of R n , and : U ∞ R be a measureable function which is bounded on every compact subset of U . For every hypercube x, r ) = { x = [ x 1 ; x 2 ; . . . ; x n ] : | x k − ¯ Q (¯ x k | ∀ r } contained in U , its weighted volume with respect to is defined by x n − 1 + r ¯ x 1 + r x 2 + r ¯ x n + r ¯ ¯ V ( Q (¯ x, r )) = . . . ( x 1 , x 2 , . . . , x n ) dx n dx n − 1 . . . dx 2 dx 1 . ¯ ¯ ¯ ¯ x 1 − r x 2 − r x n − 1 − r x n − r Without going into the fine details of the measure theory, let us say that the weighted volume of a subset X U with respect to is well defined if there exists M > such that for every > there exist (countable) families of cubes { Q 1 k } (all contained in k } and { Q 2 1 Version of October 31, 2003 2 U ) such that X is contained in the union of Q i k , the union of all Q 2 is contained in the k union of X and Q 1 k , and k ) < , V | | ( Q 2 V | | ( Q 1 k ) < M, k k in which case the volume V ( X ) is (uniquely) defined as the limit of V ( Q 2 k ) k as and Q 2 k are required to have empty pair-wise intersections. A common alternative notation for V ( X ) is V ( X ) = ( x ) dx....
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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.

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lec11_6243_2003 - Massachusetts Institute of Technology...

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