{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# A formal study of sensitivity would lead us 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: 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...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern