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

where f:R\×Rn×R→ R\ and g:R\×R\×R→ R are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory co(t), yo(t), and that the derivative

This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems 6.1 Stability of an equilibria

In particular, when o=0, this yields the definition of a Lyapunov function Finding, for a given supply rate, a valid storage function(or at least proving that one exists)is a major challenge in constructive analysis of nonlinear systems. The most com-

Definition A real-valued function V: X H R defined on state space X of a system with behavior set B and state r:B×[0,∞)→ X is called a Lyapunov function if tHv(t)=v(a(t))=v(a(z(), t)) is a non-increasing function of time for every z E B according to this definition, Lyapunov

he variable t is usually referred to as the\time Note the use of an integral form in the formal definition(2.2): it assumes that the function tHa(a(t), t)is integrable on T, but does not require =a(t)to be differentiable at any particular point, which turns out to be convenient for working with discontinuous

9. Aldehydes and Ketones Nucleophilic Addition Reactions

What is an alkyl halide An organic compound containing at least one carbon-halogen bond (C-X) X(F, Cl, Br, Replaces h Can contain many C-X bonds

5. Aromatic Compounds p: 153 Aromatic was used to described some fragrant compounds in early 19th century Not correct: later they are grouped by chemical behavior(unsaturated compounds that undergo substitution rather than addition Current: distinguished from aliphatic compounds by electronic configuration

p.91 3.1 Naming alkenes 1. Name the parent hydrocarbon, find the longest carbon chain that contains the double bond.命名母体