Search Results (137)

Consider a class of coupled discrete-time Riccati equations arising from jump systems. To compute their solutions when systems reach a steady state, we propose an operator Newton method and correspondingly establish its quadratic convergence under suitable assumptions. The advantage of the proposed method lies in the fact that its subproblems are solved using the operator Smith method, which allows it to maintain quadratic convergence in both the inner and outer iterations. Moreover, it does not require the constant term matrix of the equation to be invertible, making it more broadly applicable than existing inverse-free iterative methods. For large-scale problems, we develop a low-rank variant by incorporating truncation and compression techniques into the operator Newton framework. A complexity analysis is also provided to assess its scalability. Numerical experiments demonstrate that the presented low-rank operator Newton method is highly effective in approximating solutions to large-scale structured coupled Riccati equations. Full article

In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing $\beta$-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article

This paper addresses the problem of optimal stabilization for stochastic dynamical systems characterized by Markov switches and concentration points of jumps, which is a scenario not adequately covered by classical stability conditions. Unlike traditional approaches requiring a strictly positive minimal interval between jumps, we allow jump moments to accumulate at a finite point. Utilizing Lyapunov function methods, we derive sufficient conditions for exponential stability in the mean square and asymptotic stability in probability. We provide explicit constructions of Lyapunov functions adapted to scenarios with jump concentration points and develop conditions under which these functions ensure system stability. For linear stochastic differential equations, the stabilization problem is further simplified to solving a system of Riccati-type matrix equations. This work provides essential theoretical foundations and practical methodologies for stabilizing complex stochastic systems that feature concentration points, expanding the applicability of optimal control theory. Full article

In this contribution, a model order reduction (MOR) strategy for systems characterized by Warburg-type impedance behavior, frequently encountered in electrochemical applications, is addressed. In particular, the interest is focused on the time-domain approach for deriving low-order models of such a system, in contrast to the current approaches based on the frequency domain. By exploiting the peculiar structure of positive real (PR) systems, a characteristic value technique relying on the Riccati Equation Balancing strategy is introduced to approximate such models with reduced complexity. The characteristic values of the system are used to define suitable reduced-order models. A numerical case study is presented to validate the effectiveness of the proposed method. The model is also compared against experimental data from the literature, confirming its capability to capture dominant Warburg behavior. Performance indices are computed to quantitatively assess the approximation accuracy across different model orders. The results are critically compared with those obtained using conventional MOR techniques, allowing a thorough assessment of accuracy, stability, and implementation feasibility. Full article

Inertial instability is a key process in the dynamics of rotating and stratified fluids, which arises when the absolute vorticity of the flow becomes negative. This study explored the nonlinear behavior of inertial instability by incorporating a hidden symmetry into the equations of motion governing atmospheric dynamics. The atmosphere was modeled as a complex system composed of interacting structural elements, each capable of oscillatory motion influenced by planetary rotation and geostrophic shear. By applying a symmetry-based framework rooted in projective geometry and Riccati-type transformations, we show that synchronization and structural coherence can emerge spontaneously, independent of external forcing. This hidden symmetry leads to rich dynamical behavior, including phase coupling, quasi-periodicity, and bifurcations. Our results suggest that inertial instability, beyond its classical linear interpretation, may play a significant role in organizing large-scale atmospheric patterns through internal geometric constraints. Full article

This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. Full article

This paper introduces Disturbance-Aware Redundant Control (DARC), a control framework addressing the challenge of human–robot co-transportation under disturbances. Our method integrates a disturbance-aware Model Predictive Control (MPC) framework with a proactive pose optimization mechanism. The robotic system, comprising a mobile base and a manipulator arm, compensates for uncertain human behaviors and internal actuation noise through a two-step iterative process. At each planning horizon, a candidate set of feasible joint configurations is generated using a Conditional Variational Autoencoder (CVAE). From this set, one configuration is selected by minimizing an estimated control cost computed via a disturbance-aware Discrete Algebraic Riccati Equation (DARE), which also provides the optimal control inputs for both the mobile base and the manipulator arm. We derive the disturbance-aware DARE and validate DARC with simulated experiments with a Fetch robot. Evaluations across various trajectories and disturbance levels demonstrate that our proposed DARC framework outperforms baseline algorithms that lack disturbance modeling, pose optimization, or both. Full article

This paper studies the problem of time-varying formation-tracking control for a class of nonlinear multi-agent systems. A distributed adaptive controller that avoids the global non-zero minimum eigenvalue is designed for heterogeneous systems in which leaders and followers contain different nonlinear terms, and which relies only on the relative errors between adjacent agents. By adopting the Riccati inequality method, the adaptive adjustment factor in the controller is designed to solve the problem of automatically adjusting relative errors based solely on local information. Unlike existing research on time-varying formations with fixed and switching topologies, the method of jointly connected topological graphs is adopted to enable nonlinear followers to track the trajectories of leaders with different nonlinear terms and simultaneously achieve the control objective of the desired time-varying formation. The stability of the system under the jointly connected graph is proved by the Lyapunov stability proof method. Finally, numerical simulation experiments confirm the effectiveness of the proposed control method. Full article

This paper addresses the problem of optimal $H_2$-filtering for a class of continuous-time linear McKean–Vlasov stochastic differential equations under sampled measurements. The main tool used to solve the filtering problem is a forward jump matrix linear differential equation with a Riccati-type jumping operator. More specifically, the stabilizing solution of such a jump Riccati-type equation plays a key role. Full article

This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically. Full article

This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough–Dodd, Toda, KdV, pKdV, NLS, and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of nonlinear physics—such as Bose–Einstein condensation, superconductivity, gravity models, optics, and soliton turbulence—these results may have far-reaching applications. Full article

Quad-tilt-wing (QTW) Unpiloted Aerial Vehicles (UAVs) combine the vertical takeoff and landing capabilities of rotary-wing designs with the high-speed, long-range performance of fixed-wing aircraft, offering significant advantages in both civil and military applications. The unique configuration of QTW UAVs presents complex control challenges due to nonlinear dynamics, strong coupling between translational and rotational motions, and significant variations in aerodynamic characteristics during transitions between flight modes. To address these challenges, this study develops an optimal control framework tailored for real-time operations. A State-Dependent Riccati Equation (SDRE) approach is employed for attitude control, addressing nonlinearities, while a Linear Quadratic Regulator (LQR) is used for position and velocity control to achieve robustness and optimal performance. By integrating these strategies and utilizing the inverse dynamics approach, the proposed control system ensures stable and efficient operation. This work provides a solution to the optimal control complexities of QTW UAVs, advancing their applicability in demanding and dynamic operational environments. Full article

This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises from the need for a more generalized framework to analyze oscillatory behavior in fractional differential equations, bridging the gap in the existing literature. By employing the generalized Riccati technique and the integral averaging method, we establish new oscillation criteria that extend and refine previous results. Illustrative examples are provided to validate the theoretical findings and highlight the effectiveness of the proposed methods. Full article

We propose a wavelet collocation method for solving the fractional Riccati equation, using the Müntz–Legendre wavelet basis and its associated operational matrix of fractional integration. The fractional Riccati equation is first transformed into a Volterra integral equation with a weakly singular kernel. By employing the collocation method along with the operational matrix, we reduce the problem to a system of nonlinear algebraic equations, which is then solved using Newton–Raphson’s iterative procedure. The error estimate of the proposed method is analyzed, and numerical simulations are conducted to demonstrate its accuracy and efficiency. The obtained results are compared with existing approaches from the literature, highlighting the advantages of our method in terms of accuracy and computational performance. Full article

This paper introduces a nonlinear Lyapunov-based Finite-Time State-Dependent Differential Riccati Equation (FT-SDDRE) control scheme, considering actuator saturation constraints and ensuring that the control system operates within safe operational limits designed for satellite reconfiguration and formation-keeping in low Earth orbit (LEO) missions. This control approach addresses the challenges of reaching the relative position and velocity vectors within a defined timeframe amid various orbital perturbations. The proposed approach guarantees precise formation control by utilizing a high-fidelity relative motion model that incorporates all zonal harmonics and atmospheric drag, which are the primary environmental disturbances in LEO. Additionally, the article presents an optimization methodology to determine the most efficient State-Dependent Coefficient (SDC) form regarding fuel consumption. This optimization process minimizes energy usage through a hybrid genetic algorithm and simulated annealing (HGASA), resulting in improved performance. In addition, this paper includes a sensitivity analysis to identify the optimized SDC parameterization for different satellite reconfiguration maneuvers. These maneuvers encompass radial, along-track, and cross-track adjustments, each with varying baseline distances. The analysis provides insights into how different parameterizations affect reconfiguration performance, ensuring precise and efficient control for each type of maneuver. The finite-time controller proposed here is benchmarked against other forms of SDRE controllers, showing reduced error margins. To further assess the control system’s effectiveness, an input saturation constraint is integrated, ensuring that the control system operates within safe operational limits, ultimately leading to the successful execution of the mission. Full article