# Feedback Stabilization of First Order Neutral Delay Systems Using the Lambert W Function
## 3 Main Results  
### Oscillation and Non-Oscillation Criteria  
In this section, we discuss the oscillation and non-oscillation criteria for Equation (4). The non-oscillation criteria indicate the definite existence of some real roots. **Lemma 1**. : _If a \(<0\), then the following are true:_  
_(i)_ _The global minimum value of_ \(f(\lambda)\) _is nonpositive._  
_(ii)_ \(f(\lambda)\) _is strictly decreasing on_ \((-\infty,\lambda_{m})\)_, strictly increasing on_ \((\lambda_{m},\infty)\) _and has a global minimum point at_ \(\lambda_{m}\)_._  
**Proof. ** In Equation (5), \(f(0)=0\) and the extreme values of \(f(\lambda)\) are obtained by applying [PERSON] rule. Subsequently, \(f(-\infty)=\infty\), and \(f(\infty)=\infty\). Hence, the graph of Equation (5) opens upwards. Therefore, the global minimum can be found by setting its first derivative to zero. \[f^{\prime}(\lambda)=1+ae^{-\tau_{1}\lambda}-a\tau_{1}\lambda e^{-\tau_{1} \lambda}=0. \tag{9}\]  
Multiplying both sides of Equation (9) by \(\frac{\epsilon}{a}\), we get:  
\[(1-\tau_{1}\lambda)e^{1-\tau_{1}\lambda}=-ea^{-1}. \tag{10}\]  
Using the Lambert W function, the above equation can be rewritten as:  
\[W(-ea^{-1})e^{W(-a^{-1})}=-ea^{-1}. \tag{11}\]From Equations (10) and (11) we obtain:  
\[1-\tau_{1}\lambda=W(-ea^{-1}). \tag{12}\]  
Hence, the zero (denoted by \(\lambda_{m}\)) of Equation (9) can be written in terms of the Lambert W function as:  
\[\lambda_{m}=\frac{1}{\tau_{1}}\Big{(}1-W(-ea^{-1})\Big{)}. \tag{13}\]  
As \(a<0\), \(\lambda_{m}\) is a real value. When \(a=-1\), then \(\lambda_{m}=0\) and \(f(0)=0\). Substituting Equation (13) into Equation (5), we obtain the following equation:  
\[ \begin{array}{c}f(\lambda_{m})=\frac{1}{\tau_{1}}\Big{(}1-W(-ea^{-1})\Big{)} +\frac{1}{\tau_{1}}\Big{(}1+ae^{W(-ea^{-1})-1}\Big{)},\\ =\frac{1}{\tau_{1}}\Big{(}2-W(-ea^{-1})+ae^{W(-ea^{-1})-1}\Big{)}.\end{array}  \tag{14}\]  
To simplify, let:  
\[\gamma=2-W(-ea^{-1})+ae^{W(-ea^{-1})-1}. \tag{15}\]  
Multiplying both sides of Equation (15) by \(W(-ea^{-1})\), we obtain:  
\[\gamma W(-ea^{-1})=2W(-ea^{-1})-W^{2}(-ea^{-1})+aW(-ea^{-1})e^{W(-ea^{-1})}e^ {-1}. \tag{16}\]  
From Equation (1), \(W(-ea^{-1})e^{W(-ea^{-1})  
**Proof. ** The second derivative of \(f(\lambda)\) is given by:  
\[f^{\prime\prime}(\lambda)=-\tau_{1}ae^{-\tau_{1}\lambda}-a\tau_{1}e^{-\tau_{1} \lambda}\Big{(}1-\tau_{1}\lambda\Big{)}. \tag{22}\]  
Because \(a<0\), then \(f^{\prime\prime}(\lambda)>0\) if \(\lambda\leq\frac{1}{\tau_{1}}\). **Lemma 3. **_If \(a<-1\) and \(\lambda\leq\frac{1}{\tau_{1}}\), then \(f(\lambda)\) is convex on the interval \([0,\ \frac{1}{\tau_{1}}]\)._  
**Proof. ** The proof is similar to that of Lemma 2. **Remark 1. **_The above lemmas show that \(f(\lambda)\) is not a convex function on \(\mathrm{Re}(\lambda)\). As \(g(\lambda)\) is infinitely differentiable and its second derivative is positive, \(g(\lambda)\) is a concave function for all \(\mathrm{Re}(\lambda)\)._  
The solutions of Equation (4) are the intersection points (or tangent points) of the functions \(f(\lambda)\) and \(g(\lambda)\). For \(-1<a<0\), \(f(\lambda)<0\) for all \(\lambda\in\Big{(}\frac{1}{\tau_{1}}\ln(-a),0\Big{)}\) and \(g(\lambda)\) is a concave function. If the global minimum value, \(f(\lambda_{m})\), is greater than \(-b=\sup_{\lambda\in(-\infty,0)}g(\lambda)\), then they do not intersect at any point. This implies that Equation (3) has no real solution. **Theorem 1. **_If \(-1<a<0\) and_  
\[2-W(-ea^{-1})+ae^{W(-\alpha^{-1})-1}+b\tau_{1}>0, \tag{23}\]  
_then every solution of Equation (3) oscillates._  
**Proof. ** We obtain from Equation (14), in several steps, the following mutually equivalent inequalities:  
\[f(\lambda_{m})>-b\Leftrightarrow\ \frac{1}{\tau_{1}}\Big{(}2-W(-ea^{-1})+ ae^{W(-\alpha^{-1})-1}\Big{)}>-b,\]  
\[\Leftrightarrow\ 2-W(-ea^{-1})+ae^{W(-\alpha^{-1})-1}>-b\tau_{1},\]  
\[\Leftrightarrow\ 2-W(-ea^{-1})+ae^{W(-\alpha^{-1})-1}+b\tau_{1}>0.\]  
This completes the proof. **Theorem 2**[24] (Theorem 2).