Novel Double Integral Inequalities And Their Application To Stability Of Delayed Systems

Integral inequalities play an important role in the stability analysis for systems with time-varying delay. In this paper, the orthogonal polynomials of one variable are extended to the orthogonal system of bivariate polynomials. An orthogonal system of bivariate functions which need not be continuous is introduced by triangulating a bounded domain in the plane. The bivariate functions in this orthogonal system need not be polynomials. Based on the orthogonal decomposition of vector and orthogonal approximation of vector, some new double integral inequalities are obtained. These double integral inequalities can provide tighter bounds than most of existing inequalities. Based on these double integral inequalities, an improved sufficient condition on asymptotical stability for systems with time-varying delay is obtained. Several numerical examples are given to show the effectiveness of the stability condition proposed in this paper.