## Appendix B Proof of Corollary 1  
Assume that the stability condition of Lemma 3 is satisfied, i.e., \(\tau<\frac{1}{a}-\frac{1}{b_{1}}+\sqrt{\frac{1}{a^{2}}+\frac{1}{b_{1}^{2}}}\). Then, in order to ensure the stability of the system (1) with the injection scheme presented in Figure 1, it is required to obtain the values of the parameters \(g_{1}\) and \(g_{2}\) of the system (1). First, the value of \(g_{1}\) is calculated. Consider a root locus of the fictitious system (1) together with the proportional action \(R_{3}(z)=U_{3}(z)-g_{1}Y_{3}(z)\) (this can be seen in Figure 1), where the parameter that varies is \(g_{1}\). Since \(g_{1}\) varies from zero to positive values, the roots move in such a way that there is a breakpoint between the pole located inside the unit circle and the pole located outside it, which is calculated in Equation (1). Extending the analysis to the continuous case, in this case, the breaking point tends to one from the left, as can be seen in Equation (1). Therefore, it is possible to obtain the value of \(g_{1}\) by solving for \(g_{1}\) in the characteristic equation of the fictitious system (1) taking \(z=1\) and \(\lim_{n\to\infty}g_{1}\). This procedure can be carried out as follows. First, the value of \(g_{1}\) is given by  
\[g_{1}=\frac{\mathrm{e}^{-b_{1}T}\Big{(}-b_{1}\Big{(}\mathrm{e}^{b_{1}T}z-1 \Big{)}\big{(}az^{n}+anz^{n-1}\big{(}z-\mathrm{e}^{aT}\big{)}\big{)}-ab_{1} \mathrm{e}^{b_{1}T}z^{n}\big{(}z-\mathrm{e}^{aT}\big{)}\Big{)}}{\alpha_{1}bb_ {1}}, \tag{1}\]  
taking \(z=1\), one obtains  
\[g_{1}=\frac{a\big{(}\mathrm{e}^{aT}-1\big{)}-\mathrm{e}^{-b_{1}T}\Big{(} \mathrm{e}^{b_{1}T}-1\Big{)}\big{(}a-a\big{(}\mathrm{e}^{aT}-1\big{)}n\big{)} }{\alpha_{1}b}, \tag{2}\]  
and taking the \(\lim_{n\to\infty}g_{1}\), it is found that  
\[\lim_{n\to\infty}g_{1}=\frac{b_{1}(a\tau-1)+a}{b}. \tag{3}\]  
Since \(\lim_{n\to\infty}g_{1}=(b_{1}(a\tau-1)+a)/b\), the value of \(g_{1}\) with the trajectories and/or poles inside of the unit circle has to be greater than this value, therefore the lower bound of \(g_{1}\) is set as  
\[\frac{b_{1}(a\tau-1)+a}{b}<g_{1}<\frac{b_{1}(a\tau-1)+a}{b}+\epsilon_{1}, \tag{4}\]  
for some \(\epsilon_{1}>0\).  
Finally, to obtain the parameter \(g_{2}\), a similar procedure is performed. Consider a diagram of the locus of the roots associated with the closed-loop system (1). It is possible to obtain the value of the parameter \(g_{2}\) by solving for \(g_{2}\) from the characteristic equation of the system (1) taking \(z=1\) and by applying \(\lim_{n\to\infty}g_{2}\). The procedure is developed as follows. First, the value of \(g_{2}\) is  
\[g_{2}=-\frac{b_{1}\Big{(}\mathrm{e}^{b_{1}T}z-1\Big{)}\big{(}az^{n}\big{(}z- \mathrm{e}^{aT}\big{)}+\alpha_{1}bg_{1}\big{)}}{\alpha_{1}\alpha_{2}b}, \tag{5}\]  
with \(z=1\),  
\[g_{2}=\frac{b_{1}\Big{(}1-\mathrm{e}^{b_{1}T}\Big{)}\big{(}\alpha_{1}bg_{1}-a \mathrm{e}^{aT}+a\big{)}}{\alpha_{1}\alpha_{2}b}, \tag{6}\]  
and considering \(\lim_{n\to\infty}g_{2}\), one obtains  
\[g_{2}=\frac{b_{1}(a-bg_{1})}{b}. \tag{7}\]Performing the same analysis as that used to obtain \(g_{1}\), the value of \(g_{2}\) is given by  
\[\frac{b_{1}(a-bg_{1})}{b}<g_{2}<\frac{b_{1}(a-bg_{1})}{b}+\epsilon_{2}, \tag{10}\]  
for some \(\epsilon_{2}>0\).  
The values of \(g_{1}\) and \(g_{2}\) ((A14) and (10)) stabilize the injection scheme shown in Figure 1.