Global Finite Time Research Articles

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

On a thermo-viscoelastic system with variable exponent sources

In this paper, we study the following nonlinear thermo-viscoelastic system utt−Δu+∫0tg(t−s)Δu(s)ds+v=|u|p(x)−2u,vt−Δv=ut,in a bounded domain Ω⊂Rn. By employing the modified potential well theory, we discuss the properties of global existence and finite time blow-up of the solutions of the problem. Moreover, in the former case by constructing a suitable Lyapunov functional we prove the exponential decay for solutions and in the latter one we give the lower and upper bounds for blow-up time of the blow-up solutions.

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.

An Adaptive Prescribed Performance Tracking Motion Control Methodology for Robotic Manipulators with Global Finite-Time Stability.

In this paper, the problem of an APPTMC for manipulators is investigated. During the robot’s operation, the error states should be kept within an outlined range to ensure a steady-state and dynamic attitude. Firstly, we propose the modified PPFs. Afterward, a series of transformed errors is used to convert “constrained” systems into equivalent “unconstrained” ones, to facilitate control design. The modified PPFs ensure position tracking errors are managed in a pre-designed performance domain. Especially, the SSE boundaries will be symmetrical to zero, so when the transformed error is zero, the tracking error will be as well. Secondly, a modified NISMS based on the transformed errors allows for determining the highest acceptable range of the tracking errors in the steady-state, finite-time convergence index, and singularity elimination. Thirdly, a fixed-time USOSMO is proposed to directly estimate the lumped uncertainty. Fourthly, an ASTwCL is applied to deal with observer output errors and chattering. Finally, an observer-based-control solution is synthesized from the above techniques to achieve PCP in the sense of finite-time Lyapunov stability. In addition, the precision, robustness, as well as harmful chattering reduction of the proposed APPTMC are improved significantly. The Lyapunov theory is used to analyze the stability of closed-loop systems. Throughout simulations, the proposed PPTMC has been shown to perform well and be effective.

Global well-posedness of fourth-order Petrovsky equation with weak and strong damping terms

In this paper, a semilinear fourth-order Petrovsky equation with nonlinear weak damping term and linear strong damping term is considered in the frame of the potential well. We prove the existence and uniqueness of the global strong solution with sub-critical initial energy. In addition, we obtain the global existence, asymptotic behaviour and finite time blowup for the weak (non-steady state) solution with a critical initial energy level.

Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term

Abstract This article is devoted to study the existence of global solutions and finite time blow-up of local solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term. By applying the potential well and energy estimation method, in low initial energy and critical initial energy, we derive some sufficient conditions which are global existence and explosion of the solutions for this type of Klein-Gordon equation.

Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping

This paper deals with a class of quasilinear wave equations with structural damping or strong damping. By virtue of the improved Faedo–Galerkin method and some technical efforts, we first establish the local well-posedness of weak solutions, especially the continuity of weak solutions with respect to time in the natural phase space. Then we investigate the global existence, asymptotic behavior and finite time blow-up of weak solutions with subcritical or critical initial energy. As for the supercritical initial energy case, we show that the weak solutions may blow up in finite time with arbitrarily high initial energy. Last but not least, the upper and lower bounds of the blow-up time for blow-up solutions are derived.

Dual-objective torque control of electric vehicles drivetrain with robust global finite-time stability guarantee

The low-frequency torsional vibration caused by a sudden input torque change seriously affects electric vehicles' comfort and dynamic performance (EVs). This paper proposes a novel torque controller with robust global finite-time stability based on terminal sliding mode control to achieve the dual objectives of anti-jerk control and traction torque control. For this purpose, a general second-order system of EVs drivetrain with a disturbance term is established, and a new finite-time terminal sliding surface is designed. The proposed finite time torque controller (FT-TC) includes a finite-time sliding mode disturbance observer for real-time disturbance estimation and a finite-time sliding mode motor controller for calculating motor torque control law. In addition, the elimination of inherent chattering of the controller is considered in this paper. Finally, the proposed FT-TC is verified and compared with parallel double-loop PID controller and sliding mode controller on the Matlab/Simulink platform and an electric SUV platform under the condition of suddenly pressing/releasing the accelerator pedal during starting. The results indicate that the proposed FT-TC is effective and has a better synthetic evaluation performance in achieving the dual objectives.

Global Existence and Finite Time Blow-Up in a New Class of Non-Linear Viscoelastic Wave Equation

Global Existence and Finite Time Blow-Up in a New Class of Non-Linear Viscoelastic Wave Equation

Global existence and finite time blow-up for a stochastic non-local reaction-diffusion equation

In this paper, we investigate the existence and blow-up of positive solutions for a stochastic non-local reaction-diffusion equation. Firstly, we give the conditions for the existence of global positive solutions and estimate the probability of existence of non-trivial positive global solution. Secondly, by choosing some special initial data, we prove that stochastic non-local reaction-diffusion equation blows up in finite time in the point-wise sense with probability 1. The random upper bounds for blow-up times are obtained. Furthermore, by increasing the amplitude of the initial data, we can get a blow-up in any short time with a positive probability. Finally, we prove that the blow-up set is the whole region.

Gradient-based neural networks for solving periodic Sylvester matrix equations

This paper considers neural network solutions of a category of matrix equation called periodic Sylvester matrix equation (PSME), which appear in the process of periodic system analysis and design. A linear gradient-based neural network (GNN) model aimed at solving the PSME is constructed, whose state is able to converge to the unknown matrix of the equation. In order to obtain a better convergence effect, the linear GNN model is extended to a nonlinear form through the intervention of appropriate activation functions, and its convergence is proved through theoretical derivation. Furthermore, the different convergence effects presented by the model with various activation functions are also explored and analyzed, for instance, the global exponential convergence and the global finite time convergence can be realized. Finally, the numerical examples are used to confirm the validity of the proposed GNN model for solving the PSME considered in this paper as well as the superiority in terms of the convergence effect presented by the model with different activation functions.

Global existence and finite time blow‐up for the heat flow of H‐system with constant mean curvature

In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H‐system in a bounded smooth domain of . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions but also obtain the decay rate of the norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were.

Model free based finite time fault‐tolerant control of robot manipulators subject to disturbances and input saturation

AbstractThis article proposed a model free based finite time fault‐tolerant control (FTC) method of robot manipulators subject to unknown disturbances, input saturation, and actuator faults. First, an accuracy‐driven integral terminal sliding mode surface (ADITSMS) was constructed, and the finite‐time convergence and the parametric analysis were investigated. Second, based on the proposed ADITSMS, a double‐loop control framework which consists of the position loop and velocity loop was proposed, and the finite time command filtered algorithm was utilized to estimate the derivative of the virtual control law of the position loop, thus a novel finite‐time robust adaptive FTC was proposed. Third, to eliminate the negative effects of the faults and disturbances and implement model free control, the uncertainty and disturbance estimation is designed, and a novel adaptive algorithm is developed to suppress the estimation error whose upper bound or derivative was not required. Meanwhile, an auxiliary system is constructed to deal with the input saturation. The proposed controller possess the superiorities of model free nature and global finite time convergence property. Finally, numerical simulations and comparison studies were conducted to demonstrate the effectiveness of the proposed controller.

On a 1D model with nonlocal interactions and mass concentrations: an analytical-numerical approach* *Abreu was partially supported by CNPq 306385/2019-8, BR and Ferreira was partially supported by FAPESP and CNPq, BR.

We study global well-posedness and finite time blow-up of solutions for a nonlinear one-dimensional transport equation with nonlocal velocity , ν > 0, and measure initial data. Such model arises in fluid mechanics in vortex-sheet problems and its nonlocal feature comes from the presence of a singular integral operator (Hilbert transform ) in the velocity field. In the viscous case ν > 0, we analytically obtain an explicit condition on the size of the initial data for the global well-posedness in the framework of pseudomeasure spaces. In fact, we can give the condition depending on the initial-mass and analyze how the flow evolves from singular measures. Also, we numerically study blow-up of concentration type and global diffusion-smooth behavior of solutions. We obtain numerics that indicate the threshold value 2π for the initial-data mass that decides between blow-up or global smoothness of solutions. Such value is the same obtained for regular initial-data and by means of entropy methods. Thus, it seems to be intrinsic to the nonlocal PDE and independent of a particular framework, approach and initial-data regularity. The inviscid case ν = 0 is remarkable: simulations for model , evidence that the solution presents blow-up of concentration type with mass-preserving, while an attenuation effect is observed for the model with opposite sign in the nonlinearity , for any nontrivial (positive) measure as initial data.

Global finite-time set stabilization of spacecraft attitude with disturbances using second-order sliding mode control

The performance of attitude stabilization control algorithms for rigid spacecraft can be limited by disturbances. In this paper, the global finite-time attitude stabilization problem with disturbances is investigated and handled by constructing a second-order sliding mode controller. Firstly, a virtual controller based on set stabilization idea is constructed to globally finite-time stabilize the system. Then, a relay polynomial second-order sliding mode controller is constructed to guarantee that the tracking error toward the virtual controller will converge to zero in finite-time. Finite-time Lyapunov theory is applied to support the proof and stability analysis. The global finite-time stability holds even with bounded disturbances. The effectiveness and feasibility of the controller are illustrated by the numerical simulations.

A sharp critical threshold for a traffic flow model with look-ahead dynamics

We study a nonlocal traffic flow model with an Arrhenius type look-ahead interaction. We show a sharp critical threshold condition on the initial data which distinguishes the global smooth solutions and finite time wave break-down.

Global existence and blow-up for semilinear parabolic equation with critical exponent in R N

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].

Robust finite-time stability of nonlinear systems involving hybrid impulses with application to sliding-mode control.

This paper is concerned with the robust finite-time stability and stabilization of impulsive systems subject to hybrid disturbances that consists of external disturbances and hybrid impulses with time-varying jump maps. First, the global finite-time stability and local finite-time stability of a scalar impulsive system are ensured by the analysis of cumulative effect of hybrid impulses. Then, asymptotic stabilization and finite-time stabilization of second-order system subject to hybrid disturbances are achieved by linear sliding-mode control and non-singular terminal sliding-mode control. It shows that the stable systems under control are robust to external disturbances and hybrid impulses with non-destabilizing cumulative effect. If the hybrid impulses have destabilizing cumulative effect, the systems are also capable of absorbing the hybrid impulsive disturbances by the designed sliding-mode control strategies. Finally, the effectiveness of theoretical results is verified by numerical simulation and the tracking control of linear motor.

Long-time existence for semilinear wave equations with the inverse-square potential

In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a “fractional dimensional” wave equation and analyzing the properties of its fundamental solution, we establish a long-time existence result, for sufficiently small, spherically symmetric initial data. Together with the previously known blow-up result, we determine the critical exponent which divides the global existence and finite time blow-up. Moreover, the sharp lower bounds of the lifespan are obtained, except for certain borderline case. In addition, our technology allows us to handle an extreme case for the potential, which has hardly been discussed in literature.

Remarks on nonlinear Schrödinger equations arising on rotational Bose–Einstein condensates

In this paper, we study the Cauchy problem for nonlinear Schrödinger equations arising as an effective model of the Bose–Einstein condensate in a magnetic trap rotating with an angular velocity. We first establish sufficient conditions showing the existence of global-in-time and finite time blow-up solutions to the equation. We next derive sharp thresholds for global existence versus finite time blow-up in the mass-critical and mass-supercritical cases. We also study the existence and strong instability of ground state standing waves related to the equation. Finally we prove the existence, non-existence, and orbital stability of prescribed mass standing waves when the rotational speed is smaller than a critical value.