Consider the nonhomogeneous equation, x' = Ax + g(t), for some g(t) : R → R". Assume that A is hyperbolic, i.e it has no eigenvalues on the imaginary axis. Assume that g(t) is uniformly continuous and bounded, i.e. there exists a M > 0 such that |g(t)| < M for all t. First verify that x(t) = eA' P3v + e A(t-r) P,9(t)dr – / eA(t-7)} Pu9(T)dr, is a solution for any v E R". What is the initial condition? Now prove that the solution is bounded for all t > 0.
Consider the nonhomogeneous equation, x' = Ax + g(t), for some g(t) : R → R". Assume that A is hyperbolic, i.e it has no eigenvalues on the imaginary axis. Assume that g(t) is uniformly continuous and bounded, i.e. there exists a M > 0 such that |g(t)| < M for all t. First verify that x(t) = eA' P3v + e A(t-r) P,9(t)dr – / eA(t-7)} Pu9(T)dr, is a solution for any v E R". What is the initial condition? Now prove that the solution is bounded for all t > 0.
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I solved it but I need help in two parts.For first part, How to show thev formula is a solution of ODE?
Second, for the third part, how to show it is bounded because I can not integratw matrix?
Transcribed Image Text:Consider the nonhomogeneous equation,
x' = Ax + g(t),
for some g(t) : R → R". Assume that A is hyperbolic, i.e it has no eigenvalues on the
imaginary axis. Assume that g(t) is uniformly continuous and bounded, i.e. there exists a
M > 0 such that |g(t)| < M for all t. First verify that
rt
x(t) = e4t P,v +
T eA(t-1) Pag(7)dr - | " eA(t-r) Pug(7)dr,
is a solution for any v E R". What is the initial condition? Now prove that the solution is
bounded for all t > 0.
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