System In Two-dimensional Space Research Articles

Mild ill-posedness in L∞ for 2D magnetohydrodynamic system with damping

The present paper is devoted to the initial-value problem for the incompressible magnetohydrodynamic system in two-dimensional space where the velocity equation involves no dissipation and the vertical component equation has a damping. Here we concentrate on the stability problem for the magnetohydrodynamic system. By constructing a sequence of special initial data, we show that the solution map of vorticity is discontinuous. Our result indicates that the L∞-norm of the vorticity is mildly ill-posed.

Small solitons and multisolitons in the generalized Davey-Stewartson system

Abstract By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are constructed by stable small solitons.

Spectral Properties of the Normalized Rigidity Matrix for Triangular Formations

This work establishes properties of the normalized rigidity matrix in two- and three-dimensional spaces. The upper bound of the normalized rigidity matrix singular values is derived for minimally and infinitesimally rigid frameworks in two- and three-dimensional spaces. We prove that the transformation of a framework does not affect the normalized rigidity matrix properties. The largest minimum singular value of the normalized rigidity matrix for a rigid framework of three agents in two-dimensional space is given as well as necessary and sufficient conditions to reach that value. These results can be used in stability analysis and control design of a distance-based formation control. The numerical simulation for multi-agent systems in two-dimensional space illustrates the theoretical results. Moreover, a real-time simulation is provided to demonstrate the spectral properties of the normalized rigidity matrix.

The Optimized Path For A Mobile Robot Using Fuzzy Decision Function

Abstract The proposed paper presents the efficient obstacle avoidance mechanism in a complex environment based on the fuzzy optimized decision function. The origin point, goal point, obstacles, optimized paths, and efficient decisions mechanism are the requirements of the navigational system in two-dimensional space. During navigation, the distances measured in Euclidean Space, but the decision comprises with the proposed fuzzy function. This fuzzy function’s domain, co-domain, and range are finite. The domain lies with the set of obstacles, co-domain lies with the set of paths and the range lies with the decision. The robot follows the fuzzy rule generated by the proposed fuzzy function presented in the form of a matrix and the combination of the cell by row and column provides the resultant decision. This decision is formulated as the compact fuzzification of the two-dimensional rule over row and column of the matrix. Thus, every decision is unique and optimized. The experimental and simulational results are validated here to show the effectiveness of the proposed controller.

Disturbance‐observer‐based antiswing control of underactuated crane systems via terminal sliding mode

In this study, based on the finite-time sliding mode control method, an antiswing control law is designed for the underactuated crane systems in two-dimensional space with external disturbance. The finite-time disturbance observer is utilised to estimate the external disturbance and develop the finite-time control law. The designed controller can regulate the trolley to the planned trajectory within a finite time in the presence of external disturbance. Then it can be shown that the proposed control approach can achieve precise trolley positioning and swing suppression. Simulation results are provided to show the satisfactory control performances of the presented control method in terms of working efficiency as well as robustness with respect to external disturbance.

A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Nonlinear chaotic systems yield many interesting features related to different physical phenomena and practical applications. These systems are very sensitive to initial conditions at each time-iteration level in a numerical algorithm. In this article, we study the behavior of some nonlinear chaotic systems by a new numerical approach based on the concept of Galerkin–Petrov time-discretization formulation. Computational algorithms are derived to calculate dynamical behavior of nonlinear chaotic systems. Dynamical systems representing weather prediction model and finance model are chosen as test cases for simulation using the derived algorithms. The obtained results are compared with classical RK-4 and RK-5 methods, and an excellent agreement is achieved. The accuracy and convergence of the method are shown by comparing numerically computed results with the exact solution for two test problems derived from another nonlinear dynamical system in two-dimensional space. It is shown that the derived numerical algorithms have a great potential in dealing with the solution of nonlinear chaotic systems and thus can be utilized to delineate different features and characteristics of their solutions.

Multi-target localisation and circumnavigation by a multi-agent system with bearing measurements in 2D space

ABSTRACTThis paper investigates the problem of multi-target localisation and circumnavigation by a multi-agent system in two-dimensional space, where each target's bearing measurement can be obtained by at least one agent. Both stationary and slowly-moving targets are considered. An estimator is designed to estimate the target position; then a distributed estimation algorithm is designed for each agent to cooperatively estimate the centroid of multi-target; and finally, based on the first two steps, a circumnavigation controller is devised to guarantee that each agent circumnavigates all targets around the multi-target centroid with a desired radius. The convergence of the estimation algorithm and the stability of the circumnavigation controller are proved. A numerical simulation is provided to verify the correctness of the conclusion and the effectiveness of the proposed algorithm.

Boundary Vibration Control of Variable Length Crane Systems in Two-Dimensional Space With Output Constraints

A variable length crane system under the external disturbances and constraints is studied in the two-dimensional space. The dynamical analysis of the cable system considers the variable length, variable tension, variable speed, and the coupled vibrations of the cable in longitudinal-transverse directions. Considering output constraint problems, boundary control algorithms with output signal barriers are designed and acted on the boundary of the cable to reduce the coupled vibrations of the flexible crane cable, and to ensure the stability of the system in theory. Effectiveness and performance of the proposed control schemes are depicted via several simulation examples.

Logarithmically extended global regularity result of Lans-alpha MHD system in two-dimensional space

We study the two-dimensional generalized Lans alpha magnetohydrodynamics system. We show that the solution pairs of velocity and magnetic fields to this system preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian.

Optimal scanning control of flexible structures in two-dimensional space

A class of optimal control problems for hyperbolic systems in two-dimensional space is considered. An approach is proposed to damp the undesirable vibrations in the structures by pointwise moving force actuators extending over the spatial region occupied by the structure. A class of performance indices is introduced that includes functions of the state variable, its first and second-order space derivatives and first-order time derivative evaluated at a preassigned terminal time, and a suitable penalty term involving the control forces. A maximum principle is given for such general scanning control problem that facilitates the determination of the unique optimal control. A solution method is developed for the active vibration control of plates of general shape. The implementation of the method is presented and the effectiveness of a single moving force actuator is investigated and compared to a single fixed force actuator by a specific numerical example.

Global solutions for coupled Kuramoto-Sivashinsky-KdV system

We study the global smooth solution for the coupled Kuramoto-Sivanshinsky-KdV system in two-dimensional space. The model is proposed to describe the surface waves on multi-layered liquid films. The global solution is obtained for general initial data, using an a priori estimate for the nonlinear system, and the smoothness of such solution is established in t > 0 t >0 .

Global solution for coupled nonlinear Klein-Gordon system

The global solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.

Exact statistics of three-hard-disk system in two-dimensional space

A system with three hard particles in two-dimensional rectangle container is investigated. The system motion is chaotic and ergodic in energy surface. The partition function of the system in its energy surface is analytically calculated based on a suitable configuration classification. Theoretical results are fully confirmed by numerical simulations. A van der Waals-like behavior, i.e., increasing system size results in increasing pressure on the container wall, is revealed in some region of the container size. This generalizes the observation of the two-hard-disk system of Phys. Rev. E 65 (2002) 066104.

Trajectory optimization involving sloshing media

AbstractThis paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two‐dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non‐linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.

Trajectory optimization involving sloshing media

This paper is concerned with the optimization of the transport motion of an open topped fluid filled container within a warehouse environment. In particular, optimal trajectories of the motion of the driver–container system in two-dimensional space will be investigated via numerical solutions of the model equations using sequential quadratic programming. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is the time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The systems of ordinary and partial differential equations, representing the dynamics of the models are solved numerically using a direct shooting method. The resulting non-linear programming problem is solved using sequential quadratic programming (SQP). Copyright © 2002 John Wiley & Sons, Ltd.

SOME ALGEBRAIC SYMMETRIES OF (2, 2)-SUPERSYMMETRIC SYSTEMS

The Hilbert spaces of supersymmetric systems admit symmetries which are often related to the topology and geometry of the (target) field-space. Here, we study certain (2, 2)-supersymmetric systems in two-dimensional space–time which are closely related to superstring models. They all turn out to possess some hitherto unexploited and geometrically and topologically unobstructed symmetries, providing new tools for studying the topology and geometry of superstring target space–times, and so the dynamics of the effective field theory in these.

Ordering process and Bloch wall dynamics in a two-dimensional anisotropic spin system

The ordering process of a nonconserved anisotropic $\mathrm{XY}$-spin system in two-dimensional space is investigated. The dynamics is described by the motion of the domain wall, the so-called Bloch wall, which has chirality, in addition to the curvature of the interface. We obtain a convenient description for the dynamics of the domain wall, which has a generalized form of the Allen-Cahn-type equation of motion for the phase boundary, taking into account the effect of the width of the wall. We also discuss the short-scale behavior of the correlation function for this system when the system relaxes from an $O(2)$ symmetry state to a state having $O(1)$ symmetry.