Matrix Exponential

[gerbil]

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

 

 

[DrRocket]

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

[gerbil]

Thanks!

 

Can you refer me to a link / paper or a book that deals with the convergence of exp(A(t)) ?

 

Thanks,

Miki

[DrRocket]

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.