Caputo-Wirtinger integral inequality and its application to stability analysis of fractional-order systems with mixed time-varying delays

Various techniques of integral inequality are widely used to establish the delay-dependent conditions for the dynamics of differential systems so that the conservatism of conditions can be reduced. In the integer-order systems, the integral term of & int;???? ?pT (s)S?p(s)ds often appears in the derivative of Lyapunov-Krasovskii functional, and how to scale down this term to obtain less conservative condition is a key problem. Similarly, the integral term of & int;??(?0 ? D ?s ?p(s))T S0 ?? D ?s ?p(s)ds with fractional derivative may also be encountered in the analysis of dynamical behaviors for fractional-order systems. In view of this, the paper intends to construct several novel fractional Wirtinger integral inequalities under the sense of Caputo derivative, and to investigate the stability of fractional-order systems with mixed time-varying delays based on the constructed Caputo-Wirtinger integral inequalities. Meanwhile, in order to analyze the stability for our concerned models by the new inequalities, two new theorems of generalized fractional-order Lyapunov direct method are given. Finally, a numerical example is designed to validate the correctness and practicability of the obtained results.