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A formal study of sensitivity would lead us to the

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Unformatted text preview: system. De nition 1.2.2. The IVP (1.1.2) is well-posed if there exists positive constants k and ^ such that, for any ^, the perturbed IVP z = f (t z) + (t) z(0) = y0 + 0 0 (1.2.5) satis es jy (t) ; z (t)j k (1.2.6) whenever j 0 j < and j (t)j < for t 2 0 T ]. Again, a Lipschitz condition ensures that we are dealing with a well-posed IVP. Theorem 1.2.2. If f (t y) satis es a Lipschitz condition on f(t y) j 0 t T ;1 < y < 1g then y = f (t y ) is well posed on 0 T ] with respect to all initial data. 0 Proof. Let (t) = z(t) ; y(t) and subtract (1.1.2) from (1.2.5) to obtain 0 (t) = f (t z) ; f (t y) + (t) (0) = 0 : Taking an absolute value and using the Lipschitz condition (1.2.4) j 0 (t)j Lj (t)j + j (t)j 10 j (0)j = 0 : We easily see that j (t)j j (t)j provided that j (t)j exists and it is fairly easy to show that j (t)j exists. Additionally, by assumption, 0 0 0 0 max j (t)j < tT mtax j 0 j < T 0 0 so (t)j j 0 Lj (t)j + Multiply by the integrating factor e (e Lt j ; Lt ; j (0)j < : j (0)j < : to obtain (t)j) e 0 Lt ; Integrating L (L + 1)eLt ; 1] : Since the denominator is smallest when t = 0 j (t)j jz (t) ; y (t)j L2 = k 8t 2 0 T] thus, y = f (t y) is well posed on 0 T ]. 0 The notion of a well-posed system is related to the more common notion of stability as indicated by the following de n...
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