# Feedback Stabilization of First Order Neutral Delay Systems Using the Lambert W Function
## 1 Introduction  
Many practical problems can be described using delay differential equations (DDE), as time delays are intrinsic to many real systems. Additionally, multiple sensors, actuators, controllers, shared communication networks, and haptics introduce multiple time delays [1; 2; 3; 4]. The delays, which are either fixed constants or distributed delays, play an important role in real systems. Neutral delay differential equations (NDDE) are dynamic systems, wherein the delays appear in both the state variables and their time derivatives. Practical applications of NDDE can be seen in circuit theory [5], mechanical systems [6; 7; 8], neural networks [9], bioengineering [10], economics [11], and control engineering [12; 13; 14]. Recent research topics in the field of delay differential systems are stability analysis, bifurcation analysis, control theory and application, numerical computation, and periodic solutions analysis. Stability analysis and stabilization of neutral delay differential systems (NDDS) is a subject of great practical and theoretical importance. This field has received considerable attention over the decades because a time delay might change the performance and stability of a control system considerably. Moreover, the NDDS exhibit drastic differences in their characteristic behavior as compared to the retarded delay differential equation (RDDE), wherein the delay argument does not occur in the highest order derivative term. Comprehensive research on this subject can be found in the literature [14; 15]. However, the solution of delay differential systems is complex due to the presence of exponential type transcendental terms in the system characteristic equation. Conventional approaches used for the stability analysis of NDDS are based on the Lyapunov function. These techniques seek to determine the solution of a linear matrix inequality problem. The spectrum approach used to solve the stability problem is mainly based on the solution of an eigenvalue problem. This provides a more accurate stability analysis with less computational burden. The authors in [16], have investigated the stability of the first order RDDE using the Lambert W function, which has been implemented in all major mathematical software packages (for example, MATLAB, Mathematica, Maple, and R). [PERSON] [17] has developed a time-delayed proportional control law that stabilizes the unstable second order neutral delay systems using the parameter space approach. A delayed proportional derivative (PD) controller with a single constant delay for the second-order undamped system has been presented in [18]. In the study described in [19], the authors showed that a non-descriptor type neutral time delay system could be \(\upmu\)-stabilized using derivative feedback under certain conditions. The authors in [20], have designed a proportional state feedback controller for the NDDS with multiple time-delays by the pole placement method. The radius of the essential spectrum of the neural equation solution operator was computed, and the stabilization problem of the neutral equation was reduced to a problem involving only a finite number of characteristic roots. A Lyapunov function-based stability criterion for the scalar NDDE with a single delay has been presented in [21]. In the last few decades, there has been considerable interest in deriving new sufficient conditions for the oscillation or non-oscillation of various types of first-order NDDE solutions [22; 23; 24; 25; 26]. This has been the case due to its theoretical interest and important applications. A few sufficient conditions for NDDE with constant coefficients have been presented in [22; 23; 24]. A number of authors have established a few sufficient conditions for oscillation and non-oscillation using an auxiliary solution [25; 26]. In this study, we developed a new stabilization method for first order NDDS with two time-delays.