Lyapunov–Krasovskii functionals for scalar neutral type time delay equations

In this paper a procedure for construction of complete type Lyapunov–Krasovskii functionals for a scalar neutral type time delay equation is considered. The construction of the functionals depends on the so-called Lyapunov functions which satisfy a delay equation with additional boundary conditions. It is shown that these functionals admit lower and upper quadratic bounds. Exponential estimates for solutions of the scalar neutral type time delay equations based on the Lyapunov–Krasovskii functionals are presented. A new definition of the Lyapunov function is given, and a detailed analysis of its properties is carried out.