线性时变状态方程的解

\begin{array}{r} \dot{x} = A (t) x (t) \Rightarrow x (t) = Φ (t, t_{0}) x (t_{0}) \end{array}

只有当 $A (t)$ 和 $\int_{t_{0}}^{t} A (τ) d τ$ 满足乘法可交换条件，时变系统的自由解才可以写为封闭的幂级数形式：

\begin{array}{r} Φ (t, t_{0}) = \exp [\int_{t_{0}}^{t} A (τ) d τ] x (t_{0}) \end{array}

如果不满足乘法可交换条件，只能用级数近似：

\begin{array}{r} Φ (t, t_{0}) = I + \int_{t_{0}}^{t} A (τ_{0}) d τ_{0} + \int_{t_{0}}^{t} \int_{t_{0}}^{τ_{0}} A (τ_{1}) d τ_{1} d τ_{0} + \int_{t_{0}}^{t} \int_{t_{0}}^{τ_{0}} \int_{t_{0}}^{τ_{1}} A (τ_{2}) d τ_{2} d τ_{1} d τ_{0} + \dots \end{array}

\begin{matrix} Φ (t, t) = I \\ Φ (t_{2}, t_{1}) Φ (t_{1}, t_{0}) = Φ (t_{2}, t_{0}) \\ Φ (t, t_{0}) = Φ^{- 1} (t_{0}, t) \\ \dot{Φ} (t, t_{0}) = A (t) Φ (t, t_{0}) \end{matrix}

\begin{array}{r} \dot{x} (t) = A (t) x (t) + B (t) u (t) \Rightarrow x (t) = Φ (t, t_{0}) x (t_{0}) + \int_{t_{0}}^{t} Φ (t, τ) B (τ) u (τ) d τ \end{array}