Integral Inequalities for the Analysis of Distributed Parameter Systems: A Complete Characterization Via the Least-Squares Principle

A wide variety of integral inequalities (IIs) have been developed and studiedfor the stability analysis of distributed parameter systems using the Lyapunovfunctional approach. However, no unified mathematical framework has beenproposed that could characterize the similarity and connection between theseIIs, as most of them was introduced in a dispersed manner for the analysis ofspecific types of systems. Additionally, the extent to which the generality ofthese IIs can be expanded and the optimality of their lower bounds (LBs)remains open questions. In this study, we introduce two general classes of IIsthat can generalize nearly all IIs in the literature. The integral kernels ofthe LBs of our IIs can contain an unlimited number of weighted ℒ^2functions that are linearly independent in a Lebesgue sense. Moreover, we notonly establish the equivalence relations between the LBs of our IIs, but alsodemonstrate that these LBs are guaranteed by the least squares principle,implying asymptotic convergence to the upper bound when the kernels functionsconstitutes a Schauder basis of the underlying Hilbert space. Owing to theirgeneral structures, our IIs are applicable in a variety of contexts, such asthe stability analysis of coupled PDE-ODE systems or cybernetic systems withdelay structures.