ABSTRACT
Before introducing various types of expansions for the response of nonlinear control systems, let us summarize some classical results for the solution of linear differential equations.
Let us consider the linear time-varying differential equation
x˙(t) = m∑
i=1 αi(t)Aix(t), x ∈ Rn, x(0) = x0 (40.4)
where for i = 1, . . . ,m, αi : R → R are locally Lebesgue integrable functions and Ai are constant n× n matrices. We may also write
x(t) = x0 + m∑
αi(σ)Aix(σ) dσ
From the classical Peano-Baker scheme, there exists a series solution of Equation 40.4 of the form [1]
x(t) = x0 + m∑
) Aix0 +
) AiAjx0 + · · ·
+ m∑
αi1 (σ1) . . . αik (σk)dσ1 . . . dσk
) Ai1 . . .Aikx0 + · · · . (40.5)
This series expansion was used in quantum electrodynamics [11].
