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

Outline Launch systems characteristics Launch systems selection process Spacecraft design envelope environments

Overview lecture Space Systems Engineering presented by: Prof David Miller prepared by: Olivier de Weck Revised and augmented by: Soon-Jo Chung Chart: 1 16.684 Space Systems Product Development February 13, 2001

ARGOS Overview Angular Resolution at 2=550 nm: 5 arcsec Wavelength Regime 400-700 nm FOr (Field-of-Regard): 120(full cone)

Goal: Give necessary optics background to tackle a space mission, which includes an optical payload Light .Interaction of light w/environment o Optical design fundamentals Optical performance considerations .Telescope types and CCd design interferometer types

OVERVIEW OF THE EFFECTS OF THE SPACE ENVIRONMENT Outgassing in near vacuum Atmospheric drag Chemical reactions Plasma-induced charging Radiation damage of microcircuits, solar arrays, and sensors single event upsets in digital devices Hyper-velocity impacts