A Functional-Analytic Framework for Nonlinear Adaptive Memory: Hierarchical Kernels, State-Dependent Sensitivity, and Memory-Dependent Functionals

TL;DR

This paper introduces a comprehensive functional-analytic framework for nonlinear adaptive memory, incorporating hierarchical kernels, state-dependent sensitivities, and memory-dependent functionals to model complex temporal influences.

Contribution

It develops a novel layered approach to classify memory kernels, define adaptive sensitivities, and construct memory-dependent functionals with rigorous properties.

Findings

Established fundamental properties of the framework including convergence and boundedness.

Demonstrated that discontinuous functions like indicator functions belong to the memory set.

Proved that the memory functional exceeds the supremum norm when the maximum is interior.

Abstract

This work develops a systematic functional-analytic framework for nonlinear adaptive memory, where the influence of past events depends on both elapsed time and the state values along a trajectory. The framework comprises three hierarchical layers. First, memory kernels are classified into mathematically admissible, regular (uniformly bounded, normalized, Lipschitz), and generalized (bounded variation, possibly sign-changing) classes. Second, adaptive sensitivity functions Lambda(s, f(s)) are introduced, satisfying natural conditions; a concrete construction based on historical deviation accumulation interpolates continuously between instantaneous response and history-dependent sensitivity, with an explicit Lipschitz estimate ||Lambda_f - Lambda_g||_inf <= L_Lambda ||f - g||_inf. Third, an adaptive memory-dependent functional S_{kappa, Lambda}(f) = sup_{t in I} (|f(t)| + integral_0^t Lambda(s, f(s)) kappa(t-s) |f(s)| ds) and the associated set M_{kappa, Lambda}(I) = {f : S_{kappa, Lambda}(f) < infinity} are constructed. Fundamental properties of the framework are established, including absolute convergence, measurability, uniform boundedness, positive definiteness, and comparison with the classical supremum norm. It is shown that C(I) is strictly contained in M_{kappa, Lambda}(I), with discontinuous functions (e.g., indicator functions of subintervals) belonging to the set -- capturing abrupt signal changes such as on-off switching in nonlinear systems. When the maximum of |f| is attained in the interior of the interval, a strict inequality S_{kappa, Lambda}(f) > ||f||_inf is proved, demonstrating the nontrivial contribution of the memory component.