Nonlinear Analysis: Modelling and Control
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Fundamental contractions in suprametric spaces: Analysis and applications
Full text linkIn this manuscript, we propose the notion of a strong extended s-suprametric space, a novel extension that outperforms both s-suprametric and extended suprametric spaces. It looks into the aspects of open and closed ball topologies within this structure. It also investigates the concepts of existence and uniqueness using basic contractions viz. Banach and Kannan contractions. Illustrative examples demonstrate how the strong extended s-suprametric space outperforms its extended equivalent. Our examples demonstrate the presence and distinctness of fixed points in this scenario. Furthermore, exploiting these newly launched results, the manuscript investigates the analysis of a boundary value problem, including diffusing chemical material constrained between parallel walls with related concentrations at the boundaries, taking into account supplied raw density and recognized absorbing coefficients. It also applies these insights to a nonlinear boundary value issue involving satellite web coupling in which a thin sheet joins two cylindrical spacecraft. This coupling causes nonlinearity, resulting in a separate boundary value issue influenced by radiation effects within the satellites
Dynamical investigation of the nonlinear Schrödinger equation with second-order spatiotemporal involvement of the time-conformable operator
Full text linkThe article analyzes the application of the extended hyperbolic function technique to a conformable-operator nonlinear Schrödinger equation, incorporating group velocity dispersion coefficients and second-order spatiotemporal components. The primary objective is establishing a spectrum of solutions directly pertinent to optical fibers. The extracted results, which include bright, singular, straddled, dark-bright, and dark solitons, are obtained by hyperbolic and trigonometric function-type solutions. We exhibit contour plots with two-dimensional and three-dimensional visualizations to emphasize the implication of the proposed conformable-operator nonlinear Schrödinger equation and to depict the diverse novel optical solutions. Additionally, we study the impact of the conformable operator on these solutions, employing graphical analysis to demonstrate its implications. The governing model shows potential applications in transmitting ultra-fast pulses via optical fibers
An analytical study of the time-fractional extended shallow-water wave equation in (3 + 1)-dimension with two different derivatives and their comparison
Full text linkIn ocean physics, an essential mathematical framework for examining the dynamic behavior of waves is the (3 + 1)-dimensional generalized shallow-water wave equation. This approach is driven by the growing need to incorporate nonlinear and anomalous behaviors in shallow-water wave propagation into more realistic mathematical models. This motivation is a key consideration for improving coastal hazard prediction, mitigating tsunami impacts, optimizing renewable energy extraction, and deepening our understanding of complex coastal processes. In this paper, exact solutions of the fractional generalized shallow-water wave equation are constructed using two alternative methods: the extended modified auxiliary equation mapping method and the F-expansion approach. The extended modified auxiliary equation mapping method yielded nineteen exact solutions across two main sets, while the F-expansion method produced solutions for seventeen different cases. To visualize these, 2D and 3D graphical representations have been generated for several solutions using fractional parameter values α ∈ (0; 1], including sample values such as 0.2, 0.5, 0.7, and 0.78, to illustrate how the order of the derivative affects the soliton profile. The results show that decreasing α leads to broader and smoother soliton structures. Finally, the modulation instability of the governing model is also investigated, confirming that the established results are stable
Monotone iterative sequences for positive solutions of a p-Laplacian Hadamard fractional boundary value problem on an unbounded domain
Full text linkIn this paper, we ensure the existence and uniqueness of positive solutions for a Hadamard fractional boundary value problem with the p-Laplacian operator on an unbounded domain. The problem is formulated as a nonlinear differential equation involving a fractional derivative of order ℓ ∈ (n – 1, n], n ∈ N, along with boundary conditions of the Hadamard fractional integral and derivative. Using the monotone iterative technique, we establish the existence of positive solutions by constructing monotone sequences that approach the solution. An error estimation formula is provided. An example is also discussed to illustrate the main result
Modeling the impact of detritus-based irrigation on agricultural crop production
Full text linkThe excessive use of chemical fertilizers in agricultural farms poses a serious threat to nearby water bodies, such as lakes, ponds, etc., by causing algal blooms in these water bodies and also put agricultural sustainability at risk. This study deals with the use of algae-rich pond water for irrigation, which impacts soil fertility through organic detritus. A nonlinear mathematical model is formulated to analyze the ecological and agronomic impacts of this irrigation approach. The formulation of the model takes into account that the detritus-based pond water used for irrigation initially benefits the crop growth; but once it exceeds a certain threshold, reduces the crop yield. Furthermore, the model demonstrates how nature-based solutions can be strategically integrated into agricultural system to achieve long-term resilience. The study identifies key thresholds and behavioral transitions by detecting a saddle-node, transcritical, and Hopf bifurcations within the proposed mathematical model. To support analytical findings, we conduct numerical simulations that provide a strong evidence of the agricultural ecosystem’s resilience particularly in maintaining the crop yield under the modeled irrigation conditions. These simulations underscore the potential for managing detritus density to optimize crop productivity while minimizing ecological risks. Findings of this study can support the environment-friendly irrigation policies suited to different agroecological regions. Study reveals that detritus-based irrigation promotes crop productivity up to a critical threshold of detritus input, beyond which its effects turn inhibitory. Simultaneously, it suppresses algal blooms, thereby uncovering a natural self-regulating mechanism with significant implications for sustainable irrigation practices and nutrient management policies
Exact controllability of conformable linear systems with semilinear boundary control
Full text linkIn this manuscript, we investigate the exact controllability of a class of linear systems governed by conformable fractional derivatives of order α in (0; 1] subject to semilinear boundary control in Banach spaces. We first establish the existence of mild solutions to the associated fractional Cauchy problems. We then derive sufficient conditions ensuring the exact controllability of these conformable linear systems under semilinear boundary control actions. An abstract model of an age-structured population dynamics system is provided to illustrate the applicability of the theoretical results
Observer-based distributed optimization for linear multiagent systems with external disturbances
Full text linkThis paper addresses the distributed optimization problem in linear multiagent systems (MASs) under external disturbances. Firstly, an observation system is designed by utilizing the output values of agents, which can eliminate external disturbances of system. Secondly, an event-triggered control algorithm is proposed through the gradient information of local cost functions, and its convergence is rigorously established using the Lyapunov stability and looped functional theory. This novel event-triggered protocol incorporates dwell time within the threshold function, effectively eliminating Zeno behavior. By leveraging the looped functional technique, more relaxed conditions are derived for solving the distributed optimization problem. Finally, the validity and feasibility of the proposed protocol are substantiated through numerical simulation
Relative controllability of damped fractional differential system with distributed delays and impulses
Full text linkIn this paper, we deal with the relative controllability of linear and nonlinear damped fractional differential system with distributed delays and impulses. The sufficient and necessary conditions for the relative controllability of the linear system under consideration are given by using the controllability Gramian matrix. We also prove a sufficient condition on the nonlinear term to ensure that the above system is relative controllable. Two instances are provided to verify that our theoretical results are accurate
Null controllability of Chafee–Infante equation under discrete-time point measurements
Full text linkNonlinear system is one of the main research objects in cybernetics, and it is the main theme of cybernetics in the 21st century. Recently, the control of the reaction–diffusion equation has been widely studied, but the nonlinear reaction–diffusion equation has been rarely studied. This paper will take the Chafee–Infante equation as an example, and the null controllability of this equation will be shown. We consider the null controllability for Chafee–Infante equation with point actuations subject to a known constant delay. The point measurements can be sampled in time and transmitted through a communication network with a time-varying delay. We design an observer for the future value of the state in order to compensate the input delay, then we ensure that the estimation error vanishes exponentially with a desired decay rate by using a time-varying observer gain. By constructing Lyapunov–Krasovskii functional and combining linear matrix inequalities (LIMs), we obtain the convergence conditions. We design the boundary controller and the point controller, and we conclude that both controllers can ensure the exponential stability of the closed-loop system with an arbitrary decay rate, which is smaller than that of the observers estimation error. At last, numerical example is given