Without the loss of generality, let the initial time be zero. Given the initial condition \(\mathbf{x}(0)\), our goal is to derive the solution of the following differential equation: \[ \mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t)+\mathbf{B}(t)\mathbf{u}(t) \tag{1} \] In this equation, \(\mathbf{x}(t)\in\mathbb{R}^{n}\), \(\mathbf{u}(t)\in\mathbb{R}^{m}\), \(\mathbf{A}(t)\in\mathbb{R}^{n\times n}\), \(\mathbf{B}(t)\in\mathbb{R}^{n\times m}\). The terminologies of these elements are described in the next post. Note that this is a exemplary initial value problem.
As our first step, we focus on a simpler version of Equation 1: \[ \mathbf{\dot{x}}(t) = \mathbf{A}(t)\mathbf{x}(t), ~~~~~~~ \mathbf{x}(0) :=\mathbf{x}_0 \tag{2} \]
We reformulate Equation 2 by integrating both sides from \(0\) to \(t\): \[ \mathbf{x}(t)-\mathbf{x}_0 = \int_{0}^{t}\mathbf{A}(\tau)\mathbf{x}(\tau)d\tau, ~~~~~~ \mathbf{x}(t)= \mathbf{x}_0+ \int_{0}^{t}\mathbf{A}(\tau)\mathbf{x}(\tau)d\tau \] By this reformulation, one can consider a function \(\mathbf{P}(\cdot)\), which maps a function \(\mathbf{x}(\cdot)\in C^{n}[0, T]\)