Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality

The article is concerned with the stability analysis of time-delay systems using complete-Lyapunov functionals. This class of functionals has been employed in the literature because of their nice properties. Indeed, such a functional can be built if a system with a constant time delay is asymptotically stable. Hence, several articles aim at approximating their parameters thanks to a discretization method or polynomial modeling. The interest of such approximation is the design of tractable sufficient stability conditions expressed on the Linear Matrix Inequality or the Sum of Squares setups. In the present article, we provide an alternative method based on polynomial approximation which takes advantages of the Legendre polynomials and their properties. The resulting stability conditions are scalable with respect to the degree of the Legendre polynomials and are expressed in terms of a tractable LMI.