Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jiahao Jiang

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.

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

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.

Paper Structure

This paper contains 15 sections, 10 theorems, 26 equations.

Table of Contents

  1. INTRODUCTION
  2. A Hierarchical Framework for Memory Kernels: Concepts and Estimates
  3. Mathematically Admissible Kernels: Minimal Requirements and Basic Properties
  4. Regular Admissible Kernels: Enhanced Regularity Conditions
  5. Generalized Admissible Kernels: Relaxed Regularity Conditions
  6. Stationary Memory Kernels and Their Representation
  7. Typical Examples of Kernel Functions
  8. Essential Integral Estimates for Admissible Kernels
  9. Adaptive Sensitivity Functions: Definition, Construction, and Basic Properties
  10. Mathematical Framework of Adaptive Sensitivity Functions
  11. A Constructive Example: Sensitivity with Historical Deviation Accumulation
  12. Adaptive Memory Sets: Functional Construction and Fundamental Properties
  13. Construction and Well-Posedness of Fundamental Memory Functions
  14. Definition and Basic Properties of the Adaptive Memory-Dependent Functional
  15. Construction of the Adaptive Memory-Dependent Sensitivity Set

Key Result

Lemma 2.1

Let $\kappa\in\mathscr{K}_{\mathrm{reg}}$ be a regular admissible kernel in the sense of Definition def:regular_admissible_kernel. Then the following estimates hold: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (40)

  • Definition 2.1: Mathematically admissible kernel
  • Remark 2.1
  • Remark 2.2: Stationarity as a foundational premise
  • Definition 2.2: Regular admissible kernel
  • Remark 2.3: Mathematical and physical interpretation of the regularity conditions
  • Definition 2.3: Generalized admissible kernel
  • Remark 2.4: Interpretation of the generalized conditions
  • Remark 2.5: Distinctive features and role of the generalized kernel class
  • Definition 2.4: Stationary memory kernel
  • Remark 2.6: Interpretation of the stationarity assumption
  • ...and 30 more