[Solved] Suppose A(t) is T-periodic. Prove that for every t_0 and every T-periodic function f(t), there exists x_0 such that the solution of \dotx(t)=A(t)
Question: Suppose \(A(t)\) is T-periodic. Prove that for every \(t_{0}\) and every T-periodic function \(f(t)\), there exists \(x_{0}\) such that the solution of
\[\dot{x}(t)=A(t) x(t)+f(t), x\left(t_{0}\right)=x_{0}\]is T-periodic if and only if \(f(t)\) is such that
\[\int_{t_{0}}^{t_{0}+T} z^{T}(t) f(t) d t=0\]for all T-periodic solution \(z(t)\) of the adjoint state equation
\[\dot{z}(t)=-A^{T}(t) z(t), z\left(t_{0}\right)=z_{0}\]
Deliverable: Word Document 
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