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 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 suc methods 11.1 Formulae for volume evolution This section presents the standard formulae for evolution of volumes 11.1.1 Weighted volume bounded on every compact subset of U. For every hypercube sureable function which is Let U be an open subset of R", and P: U HR be a me Q(z,r)={x={x1;x2;……;rn]:|rk-k≤r} contained in U, its weighted volume with respect to p is defined by 1+r r Ve(Q(i, r)) P( 1,T 1-T n-1-T Without going into the fine details of the measure theory, let us say that the weighted volume of a subset X CU with respect to p is well defined if there exists M>O such that for every e>0 there exist(countable) families of cubes Qk and ( Q2)(all contained in Version of October 31. 2003Massachusetts 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 Analysis1 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 Rn, and � : U ∞� R be a measureable function which is bounded on every compact subset of U. For every hypercube Q(¯x, r) = {x = [x1; x2; . . . ; xn] : |xk − x¯k| ∀ r} contained in U, its weighted volume with respect to � is defined by � x xn−1+r ¯ � � � � ¯ ¯ 1+r � x2+r � ¯ � xn+r V�(Q(¯x, r)) = . . . �(x1, x2, . . . , xn)dxn dxn−1 . . . dx2 dx1. x¯ ¯ 1−r x2−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 > 0 such that for every � > 0 there exist (countable) families of cubes {Q1 k k} and {Q } (all contained in 2 1Version of October 31, 2003