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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 proposed and studied
for the stability analysis of distributed parameter systems using the Lyapunov
functional approach. However, no unified mathematical framework has been
proposed that could characterize the similarity and connection between these
IIs, as most of them was introduced in a dispersed manner for the analysis of
specific types of systems. Additionally, the extent to which the generality of
these IIs can be expanded and the optimality of their lower bounds (LBs)
remains open questions. In this work, we present two general classes of IIs
that can generalize almost all IIs in the literature, whose integral kernels
can contain a unlimited number of weighted L2 functions that are linearly
independent in a Lebesgue sense. Moreover, we not only demonstrate the
equivalence between the LBs of the proposed IIs under the same kernels and
weighted functions, but also show that these LBs are guaranteed by the least
squares principle, implying asymptotic convergence to the upper bound when the
kernels functions constitutes a Schauder basis of the underlying Hilbert space.
Given their general structures, the proposed IIs can be applied in various
situations such as the stability analysis of coupled PDE-ODE systems or
cybernetic systems that can be characterized by delay structures.Comment: Submitted to ACC 202
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