  A wide variety of integral inequalities (IIs) have been developed 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 study, we introduce two general classes of IIs
that can generalize nearly all IIs in the literature. The integral kernels of
the LBs of our IIs can contain an unlimited number of weighted $\mathcal{L}^2$
functions that are linearly independent in a Lebesgue sense. Moreover, we not
only establish the equivalence relations between the LBs of our IIs, but also
demonstrate 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. Owing to their
general structures, our IIs are applicable in a variety of contexts, such as
the stability analysis of coupled PDE-ODE systems or cybernetic systems with
delay structures.
