{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 0 the notion of a well posed system is related to the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ition. De nition 1.2.3. Consider the di erential equation y = f (t y) and (without loss of 0 generality) let the origin (0 0) be an equilibrium point, i.e., f (0 0) = 0. Then, the origin is: stable if a perturbation of the initial condition jy(0)j < grows no larger than for subsequent times, i.e., if jy(t)j < for t > 0. asymptotically stable if it is stable and jy (0)j < implies that limt !1 jy (t)j = 0. unstable if it is not stable. Remark 3. This de nition could, like De nition 1.2.2, also involve perturbations of f (t y). We have omitted these for simplicity. 11 An autonomous system is one where f (t y) does not explicitly depend on t, i.e., f (t y) = f (y). If (0 0) is an equilibrium point then, in this case, f (0) = 0. Expanding the solution in a Taylor's series in y, we have y (t) = f (y) = f (0) + fy (0)y(t) + O(y2): 0 Since f (0) = 0, y (t) = fy (0)y(t) + O(y2) 0 where fy = @[email protected] Recall that a function g(y) is O(yp) if there exists a constant C > 0 such that jg(y)j C yp as y ! 0. Letting = fy (0...
View Full Document

{[ snackBarMessage ]}