Bessel–Laguerre inequality and its application to systems with infinite distributed delays

By taking advantage of properties of the Laguerre polynomials, we propose a new inequality called Bessel–Laguerre integral inequality, which can be applied to stability analysis of linear systems with infinite distributed delays and with general kernels. The matrix corresponding to the system without the delayed term or the matrix corresponding to the system with the zero-delay is not necessarily assumed to be non-Hurwitz. Through a Laguerre polynomials approximation of kernels, the advantage of the method is that the original system is not needed to be transformed into an augmented one. Instead, it is represented as a system with additional signals that are captured by the Bessel–Laguerre integral inequality. Then, we derive a set of sufficient stability conditions that is parameterized by the degree of the polynomials. The particular case of gamma kernel functions can be easily considered in this analysis. Numerical examples illustrate the potential improvements achieved by the presented conditions with increasing the degree of the polynomial, but at the price of numerical complexity.