Abstract: This paper provides a rigorous and systematic exposition of Lyapunov stability theory and LaSalle’s invariance principle for nonlinear autonomous dynamical systems. The analysis begins with the foundational formulation of equilibrium points and the construction of Lyapunov functions, emphasizing their roles as generalized energy measures for dynamical systems. Key structural properties, including positive definiteness and radial unboundedness, are introduced and used to establish global boundedness and stability properties of system trajectories.

The classical Lyapunov direct method is developed to establish global asymptotic stability under strict decay conditions on the Lyapunov function derivative. This framework is subsequently extended through LaSalle’s invariance principle, which relaxes the requirement of strict negativity and allows for semidefinite dissipation. In this setting, convergence is characterized in terms of the largest invariant subset of theV set where the Lyapunov derivative vanishes.

Supporting results are presented concerning monotonicity of Lyapunov functions along trajectories, compactness of sublevel sets, and invariance properties of limit sets. Particular emphasis is placed on stability conditions of the form

V(x)  ≤ – k ∥ x - x*∥2, and Vx   ≤ 0 and their implications for asymptotic behaviour.

A geometric interpretation is provided in which Lyapunov functions define a generalized energy landscape, and system trajectories evolve toward invariant sets where dissipation ceases. The results demonstrate that Lyapunov’s direct method and LaSalle’s invariance principle together form a unified and powerful analytical framework for establishing global asymptotic stability in nonlinear control systems, optimization dynamics, and dissipative mechanical systems.

Keywords: Lyapunov stability; LaSalle invariance principle; nonlinear dynamical systems; global asymptotic stability; Lyapunov function; radial unboundedness; equilibrium points; invariant sets; nonlinear control theory; stability analysis.

Title: A Robust Discuss on Lyapunov Stability and LaSalle’s Invariance Principle

Author: Ogbonna Nnamuchi

International Journal of Novel Research in Physics Chemistry & Mathematics

ISSN 2394-9651

Vol. 13, Issue 2, May 2026 - August 2026

Page No: 35-41

Novelty Journals

Website: www.noveltyjournals.com

Published Date: 29-May-2026

Abstract: Patient recovery is increasingly understood as a nonlinear adaptive process governed by biological, neurological, and therapeutic interactions. This paper develops a mathematically rigorous analysis for patient recovery using exponential stability theory, Lyapunov analysis, and nonlinear rehabilitation dynamics.

Keywords: Mathematical rehabilitation, exponential stability, dynamical systems, recovery dynamics.

Title: EXPONENTIAL STABILITY AND PATIENT RECOVERY DYNAMICS: MATHEMATICAL REHABILITATION FRAMEWORK

Author: Ogbonna Nnamuchi, Amaka Nnamuchi, Nwafia Walter Chukwuma

International Journal of Novel Research in Physics Chemistry & Mathematics

ISSN 2394-9651

Vol. 13, Issue 2, May 2026 - August 2026

Page No: 29-34

Novelty Journals

Website: www.noveltyjournals.com

Published Date: 28-May-2026

Abstract: Standard electromagnetic interference (EMI) shielding relieses on doping the insulating polymer matrix with conductive materials (e.g., multi-walled carbon nanotubes (MWCNT), copper, or graphene) to physically absorb or reflect the waves.

In contrast, AI models for MRET materials focus on biophysical neutralization and harmonic manipulation rather than physically blocking the signal. This allows the nylon to mitigate the biological risks of radiation exposure while permitting the cellular or wireless signals to pass through unimpeded. MRET nylon is a specialized polymer material that is structured at the nanoscale into fractal geometries. When subjected to external EMR (such as microwave or radio frequency signals from cellular phones), the piezoelectric properties of the polymer cause it to generate subtle, random, low-frequency oscillations.

Keywords: AI models, MRET materials, biological risks, radiation exposure.

Title: AI computational modeling of MRET -Nylon compound mitigation effect against non-ionizing electromagnetic radiation (Review)

Author: Igor Smirnov

International Journal of Novel Research in Physics Chemistry & Mathematics

ISSN 2394-9651

Vol. 13, Issue 2, May 2026 - August 2026

Page No: 19-28

Novelty Journals

Website: www.noveltyjournals.com

Published Date: 26-May-2026

Abstract: The increasing emphasis on sustainable technologies has encouraged the utilization of agricultural waste as a renewable source for the development of functional materials. Residues such as rice husk, wheat straw, sugarcane bagasse, and corn cob are rich in silica and can be converted into efficient adsorbents for wastewater remediation. This review discusses the extraction of silica from agricultural by-products using different approaches, including controlled combustion, acid–alkali treatment, and sol–gel synthesis, which significantly affect the structural, morphological, and surface properties of the final material. The obtained silica is generally amorphous and exhibits high porosity, large surface area, and abundant silanol groups, making it highly effective for the removal of heavy metals, dyes, and other hazardous pollutants from aqueous media. The adsorption process is influenced by several interactions, such as electrostatic attraction, hydrogen bonding, surface complexation, and ion exchange, depending on the nature of the adsorbent and solution conditions. Surface engineering strategies, including amine functionalization, metal oxide incorporation, and composite fabrication, have been explored to improve adsorption capacity and selectivity. Reported studies demonstrate that silica derived from agricultural residues can perform similarly to conventional commercial adsorbents while offering advantages related to low production cost, resource availability, and environmental sustainability. Although challenges associated with regeneration, consistency of raw materials, and industrial-scale implementation still need attention, biomass-derived silica represents a promising and environmentally responsible material for sustainable water treatment applications.

Keywords: Adsorption; Agricultural waste; Biosilica; Heavy metal removal; Rice husk ash; Water purification.

Title: Agricultural Waste- Derived Silica as a Sustainable Adsorbent for Water Purification: A Review

Author: Sweety Monga

International Journal of Novel Research in Physics Chemistry & Mathematics

ISSN 2394-9651

Vol. 13, Issue 2, May 2026 - August 2026

Page No: 1-18

Novelty Journals

Website: www.noveltyjournals.com

Published Date: 08-May-2026