121 Conclusion of proof

$$\parallel X(t)\parallel \le M\parallel X_0\parallel +KM{\int }_0^t{e}^{-\delta (t-s)}𝑑s$$

$$=M\parallel X_0\parallel +\frac{KM}{\delta }{\left[-e^{-\delta (t-s)}\right]}_0^t$$

$$=M\parallel X_0\parallel +\frac{KM}{\delta }\left(1-e^{-\delta t}\right)$$

$$\le M\parallel X_0\parallel +\frac{KM}{\delta }\mathit{\hspace{1em}\hspace{1em}}(t>0).$$

Hence $X(t)$ is bounded, so the output is also bounded, since

$$Y(t)=CX(t)+Du(t)$$

is the sum of two bounded functions.