Definition (Weighting pattern)

Given an LTVC system $$\begin{cases} \dot{x}(t)=A(t)x(t)+B(t)u(t) \\ y(t)=C(t)x(t) \end{cases}$$A weighting pattern, $T(t,\tau)$ defined as $$T(t,\tau)=C(t)\Phi_{A}(t,\tau)B(\tau)$$describes the relationship between input $u(t)$ and output $y(t)$ in the following form: $$y(t)=\int\limits _{t_{0}}^{t_{1}}T(t,\tau)u(\tau) \, d\tau$$