gerbil 0 Posted February 7, 2012 Hello All, My question is on the problem of solving the ODE x' = A(t) x (with x(0) = x0) and A belongs to R(NXN). During my research I've encountered the question of whether the sum exp(A(t)) = sum ( I + A(t) + 0.5*(A(t))^2 + ... ) is uniformly convergenct with respect to t, where t belongs to [0, T] and T is finite. One can assumes smooth matrix A(t) and A(t) is invertible. Moreover, it can be assumed that A(t) is also skew-symmetry if it helps, and that the limit exp(A(t)) exist and finite for all t. The question comes from the problem of differentiating exp(A(t)) term by term. ... So, is it true that "sum ( I + A(t) + 0.5*(A(t))^2 + ... ) " is uniformly convergent in t? Best, Miki 0 Share this post Link to post Share on other sites

DrRocket 337 Posted February 7, 2012 (edited) Hello All, My question is on the problem of solving the ODE x' = A(t) x (with x(0) = x0) and A belongs to R(NXN). During my research I've encountered the question of whether the sum exp(A(t)) = sum ( I + A(t) + 0.5*(A(t))^2 + ... ) is uniformly convergenct with respect to t, where t belongs to [0, T] and T is finite. One can assumes smooth matrix A(t) and A(t) is invertible. Moreover, it can be assumed that A(t) is also skew-symmetry if it helps, and that the limit exp(A(t)) exist and finite for all t. The question comes from the problem of differentiating exp(A(t)) term by term. ... So, is it true that "sum ( I + A(t) + 0.5*(A(t))^2 + ... ) " is uniformly convergent in t? Best, Miki So long as the norm of A is bounded on [0,T] (T finite), which it will be if A is a continuous function of t, you will have uniform convergence. This follows from the power series and the fact that [math] ||A(t)^n|| \le ||A(t)||^n[/math] which is true in any Banach algebra. The proof is exactly the same as that in the case that A is a scalar. Edited February 7, 2012 by DrRocket 0 Share this post Link to post Share on other sites

gerbil 0 Posted February 9, 2012 Thanks! Can you refer me to a link / paper or a book that deals with the convergence of exp(A(t)) ? Thanks, Miki 0 Share this post Link to post Share on other sites

DrRocket 337 Posted February 10, 2012 Thanks! Can you refer me to a link / paper or a book that deals with the convergence of exp(A(t)) ? Thanks, Miki Like I said, it is precisely the same proof as in the scalar case. Try any book on functional analysis. I haven't specifically checked by you ought to be able to find it in the books by Rudin or Yosida. Really any treatment of Banach algebras. But note this is tantamount to killing a fly with a sledge hammer. You just write down the power series and apply the inequality that I gave your earlier. 0 Share this post Link to post Share on other sites