Finite-time Behavior Research Articles

A mean field game approach for a class of linear quadratic discrete choice problems with congestion avoidance

This paper approaches large multi-agent dynamic discrete choice problems, such as those arising in crowd evacuation or micro-robotic-based exploration of unknown terrain, via a linear quadratic mean field games framework (LQ-MFG). Two particular features distinguish the proposed LQ-MFG: (i) agents have to reach within a fixed finite time one of a predefined set of possible destinations, which depend on their initial position; (ii) agent running costs can become negative as agents accrue a reward the further they remain from other agents aiming for the same set of destinations, so as to simulate crowd avoidance with limited surroundings awareness. The desirable tractability of the LQ-MFG setup must be balanced in this case by the fact that feature (ii) of the model can lead to agents escaping to infinity in finite time. An upper bound on the time horizon is derived to guarantee that the finite escape time behavior is avoided. The existence of a Nash equilibrium for the infinite population is then established, and it is shown that this equilibrium remains ɛ-Nash for the more realistic case of a large but finite population of agents. Finally, the model behavior is explored via simulations for different scenarios.

Asymptotic and Finite-Time Semistability for Nonlinear Discrete-Time Systems With Application to Network Consensus

This article focuses on semistability and finite-time semistability analysis and synthesis of discrete-time dynamical systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this article, we build on the theories of semistability and finite-time semistability for continuous-time dynamical systems to develop a rigorous framework for discrete semistability and discrete finite-time semistability. Specifically, Lyapunov and converse Lyapunov theorems for semistability and finite-time semistability are developed, and the regularity properties of the Lyapunov function establishing finite-time semistability are shown to be related to the settling time function capturing the finite settling time behavior of the dynamical system. These results are then used to develop a general framework for designing semistable and finite-time semistable consensus protocols for discrete dynamical networks for achieving multiagent coordination tasks asymptotically and in finite time. The proposed controller architectures involve the exchange of generalized energy state information between agents guaranteeing that the closed-loop dynamical network is semistable to an equipartitioned equilibrium representing a state of consensus consistent with basic thermodynamic principles.

Monotone iterative finite volume algorithms for coupled systems of first‐order nonlinear PDEs

AbstractThis paper studies coupled systems of first‐order nonlinear partial differential equations, where the first equation is an advection equation with a source term. The system is known to model physical phenomena such as general blood–tissue exchange (BTEX) and the gas–solid interphase heat transfer for the fast igniting catalytic converter. We propose a finite volume implicit–explicit approximation for the system and establish the existence and uniqueness of the classical solution of the system using the method of upper and lower solutions. The error estimates for the numerical scheme are also derived for each iteration of the monotone iterative method. Numerical tests show that the proposed scheme can accurately describe the behavior of various physical phenomena. The performance of the scheme is compared with the existing results in the literature and the numerical solutions are shown to preserve physical properties of the solutions such as positivity, blow up behavior in finite time, and concentration of the impulse.

A distributed adaptive architecture with the nonlinear reference model for safe finite-time control of uncertain multiagent systems

ABSTRACT We propose a new distributed adaptive control architecture for finite-time control of uncertain nonlinear multiagent systems. The proposed architecture employs three key components for each agent; a nonlinear reference model, a weight update rule, and an adaptive control signal. Predicated on agent-wise reference model state exchange, an ideal finite-time behaviour of overall multiagent system is captured by nonlinear reference models. The weight update rule of each agent, which is driven by a local error signal between the actual uncertain state of an agent and its reference model state, then adjusts agent-wise controller parameters in real-time to drive the local error signals of each agent to zero in finite-time. That is, not only the states of agents converge to their nonlinear reference model states in finite-time, but also the latter states converge to the given ideal behaviour in finite-time. Considering safety, the distinct feature of our architecture is that it does not rely on agent-wise actual state exchange between agents, which involves the effect of system uncertainties. This implies that when a subset of agents exhibits, for example, Byzantine behaviour, then their behaviour do not affect the rest of multiagent system from functioning.

Finite-time Behaviour of Solutions to Nonlinear Parabolic Equation

In this paper we consider nonlinear parabolic equation with power type nonlinearity which such types of problems occur in many mathematical models of applied science. We show that there are solutions under some conditions on initial data which blow up in finite time with positive initial energy.

Fixed time stability and optimal stabilisation of discrete autonomous systems

ABSTRACT Unlike finite time stability, wherein the upper bound of the settling-time function capturing the finite settling time behaviour of the dynamical system depends on the system initial conditions, fixed time stability involves finite time stable systems for which the minimum bound of the settling-time function is guaranteed to be independent of the system initial conditions and can a priori be adjusted. In this paper, we develop several fixed time stability results for discrete autonomous systems including a fixed-time Lyapunov theorem that involves a Lyapunov difference that satisfies an exponential inequality of the Lyapunov function giving rise to a minimum bound on the settling-time function characterised by the primary and secondary branches of the Lambert W function. Using these results, we develop an optimal control framework by exploiting connections between Lyapunov theory for fixed time stability and Bellman optimal control theory. In particular, we show that fixed time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that serves as the solution to the steady state Bellman equation guaranteeing both fixed time stability and optimality.

A Note on Fokker–Planck Equations and Graphons

Fokker-Planck equations represent a suitable description of the finite-time behavior for a large class of particle systems as the size of the population tends to infinity. Recently, the theory of graph limits has been introduced in the classical mean-field framework to account for heterogeneous interactions among particles. In many instances, such network heterogeneity is preserved in the limit which turns from being a single Fokker-Planck equation (also known as McKean-Vlasov) to an infinite system of non-linear partial differential equations (PDE) coupled by means of a graphon. While appealing from an applied viewpoint, few rigorous results exist on the graphon particle system. This note addresses such limit system focusing on the relation between interaction network and initial conditions: if the system initial datum and the graphon degrees satisfy a suitable condition, a significantly simpler representation of the solution is available. Interesting conclusions can be drawn from this result: substantially different graphons can give rise to exactly the same particle behavior, e.g., if the initial condition is independent of the graphon, the particle behavior is indistinguishable from the well-known mean-field limit as long as the graphon has constant degree; step-kernels, a building block of graph limit theory, can be used to approximate the graphon particle system with a finite system of Fokker-Planck equations and an explicit rate of convergence.

Statistical generalization of regenerative bosonic and fermionic Stirling cycles.

We have constructed a unified framework for generalizing the finite-time thermodynamic behavior of statistically distinct bosonic and fermionic Stirling cycles with regenerative characteristics. In our formalism, working fluids consisting of particles obeying Fermi-Dirac and Bose-Einstein statistics are treated on equal footing and modeled as a collection of noninteracting harmonic and fermionic oscillators. In terms of the frequency and population of the two oscillators, we have provided an interesting generalization for the definitions of heat and work that are valid for classical as well as nonclassical working fluids. Based on a generic setting under finite-time relaxation dynamics, nice results on low- and high-temperature heat transfer rates are derived. Characterized by equal power, efficiency, entropy production, cycle time, and coefficient of performance, the thermodynamic equivalence between two types of Stirling cycles is established in the low-temperature "quantum" regime.

Pseudoparabolic equations with variable exponents and coefficients: blow-up and large time behaviors

ABSTRACT In this paper, the following nonlinear generalized pseudoparabolic equation with variable exponents and coefficients is considered: Under suitable assumptions on the specified initial data and the variable exponents q, p, m, and the coefficients , the finite-time blow-up and large time behavior (exponential and power decay) of the weak solutions are established.

Technical Note—On Nested Partitions Method for Global Optimization

Finite-Time Behavior of Nested Partitions Method for Global Optimization

Isospectral Twirling and Quantum Chaos.

We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE).

Prescribed chattering reduction control for quadrotors using aperiodic signal updating

In the paper, a prescribed chattering reduction control using aperiodic signal updating is presented for quadrotors subject to parameter uncertainties and external disturbances. Using estimation errors instead of tracking errors to update adaptive laws, estimator-based minimum learning parameter (EMLP) observers capable of relaxing computational complexity are respectively explored in translational and rotational loops to reject fast time-varying disturbances, such that transient oscillations can be efficiently mitigated even with a large adaptive gain. Meanwhile, quantitative analysis for transient learning performance is characterized by means of L2 norms of time differential of neural network weights. With the aid of disturbance estimates, a relative event-triggered robust control law is derived by inserting a compensation term to guarantee a favorable trajectory tracking with Zeno free behaviors and decreased sampling cost. Besides, an appointed-time prescribed performance control (APPC) is established, enforcing trajectory tracking errors to evolve within pre-given regions even in face of triggering errors, where a piecewise and continuous finite-time behavior function, rather than an exponential decaying function, is applied to enable a preassigned fast convergence time without retuning controller parameters. Finally, the stability of closed-loop system is proved via Lyapunov synthesis, while comparative studies are provided to validate the effectiveness of presented control method.

The impact of chaotic saddles on the synchronization of complex networks of discrete-time units

A chaotic saddle is a common nonattracting chaotic set well known for generating finite-time chaotic behavior in low and high-dimensional systems. In general, dynamical systems possessing chaotic saddles in their state-space exhibit irregular behavior with duration lengths following an exponential distribution. However, when these systems are coupled into networks the chaotic saddle plays a role in the long-term dynamics by trapping network trajectories for times that are indefinitely long. This process transforms the network’s high-dimensional state-space by creating an alternative persistent desynchronized state coexisting with the completely synchronized one. Such coexistence threatens the synchronized state with vulnerability to external perturbations. We demonstrate the onset of this phenomenon in complex networks of discrete-time units in which the synchronization manifold is perturbed either in the initial instant of time or in arbitrary states of its asymptotic dynamics. The role of topological asymmetries of Erdös–Rényi and Barabási–Albert graphs are investigated. Besides, the required coupling strength for the occurrence of trapping in the chaotic saddle is unveiled.

Finite-time two-spin quantum Otto engines: Shortcuts to adiabaticity vs. irreversibility

We propose a quantum Otto cycle in a two spin-1/2 anisotropic XY model in a transverse external magnetic field. We first characterize the parameter regime that the working medium operates as an engine in the adiabatic regime. Then, we consider finite-time behavior of the engine with and without utilizing a shortcut to adiabaticity (STA) technique. STA schemes guarantee that the dynamics of a system follows the adiabatic path, at the expense of introducing an external control. We compare the performance of the nonadiabatic and STA engines for a fixed adiabatic efficiency but different parameters of the working medium. We observe that, for certain parameter regimes, the irreversibility, as measured by the efficiency lags, due to finite-time driving is so low that nonadiabatic engine performs quite close to the adiabatic engine, leaving the STA engine only marginally better than the nonadiabatic one. This suggests that by designing the working medium Hamiltonian one may spare the difficulty of dealing with an external control protocol.

Fuzzy rule-based neural appointed-time control for uncertain nonlinear systems with aperiodic samplings

In this paper, a fuzzy rule-based neural appointed-time control scheme for uncertain nonlinear systems with aperiodic samplings is proposed. To guarantee tracking properties with preassigned convergence time and remove the high dependency on accurate initial states of system, we construct an improved prescribed performance control (IPPC) scheme on the basis of a hyperbolic cosecant performance function with a finite-time behavior. In light of the unavailability of partial states and uncertainties, by combining a fuzzy wavelet neural network (FWNN) with a state observer, a FWNN-based state observer is developed, which can simultaneously approximate unavailable system states and unknown lumped disturbances with a remarkable accuracy. In addition, aimed at eliminating the problem of parameter updating explosion caused by overlarge learning dimensions, a minimum-learning-parameter (MLP) technique is embedded in the FWNN-based state observer, where the norm of weight matrix is employed for online adaptive updating. Furthermore, an event-triggered control scheme with relative thresholds is synthesized within the framework of dynamic surface control (DSC), which can allow for aperiodic samplings to save communication and actuating resources. Meanwhile, a Nussbaum type function is introduced to solve the issue of unknown control coefficients. The significant features of our work are twofold: (1) Appointed-time tracking performances with much fewer sampling times are assured. (2) An enhanced robustness against uncertainties is achieved with a lower computational complexity and the requirement of full-state measurements is relaxed. Finally, two simulation examples are performed to validate the effectiveness and advantages of the proposed control scheme.

Well-posedness and blow-up properties for the generalized Hartree equation

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

Finite horizon nonlinear optimal control for a quadrotor: Experimental results

SummaryIn this article, a suboptimal nonlinear discrete control sequence for nonlinear discrete affine control systems is proposed. Using the dynamic programming approach in discrete time domain, the suboptimal control sequence is obtained in every step considering a quadratic performance index of finite horizon. The proposed control strategy is applied to improve the dynamic and energetic performances of a quadrotor unmanned aerial vehicle, when it is subject to special maneuvers or flight conditions, such as the take off and landing phases or trajectory tracking. For these tasks, it could be required convergence in finite time and good behavior. Performing a comparative experimental study with others controllers (optimal linear and PD controllers), the effectiveness of proposed approach is proven.

Finite-time stability of discrete autonomous systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. In this paper, we address finite time stability of discrete-time dynamical systems. Specifically, we show that finite time stability leads to uniqueness of solutions in forward time. Furthermore, we provide Lyapunov and converse Lyapunov theorems for finite-time stability of discrete autonomous systems involving scalar difference fractional inequalities and minimum operators. In addition, lower semicontinuity of the settling-time function capturing the finite settling time behavior of the dynamical system is studied and illustrated through several examples. In particular, it is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov theorems for finite time stability of discrete-time systems can only assure the existence of lower semicontinuous Lyapunov functions.

Event-triggered neural intelligent control for uncertain nonlinear systems with specified-time guaranteed behaviors

In this paper, an event-triggered neural intelligent control for uncertain nonlinear systems with specified-time guaranteed behaviors is proposed. To cope with constrained communication resources, an event-triggered mechanism using switched thresholds is devised without involving input-to-state stability assumption, such that a better design flexibility and freedom can be provided. In addition, a minimum-learning-parameter-based state observer is developed to online estimate the unavailable states and uncertainties at the same time, which effectively eliminates the issue of learning explosion without sacrificing the identification precision. Furthermore, in pursuit of making a compromise between sampling cost and tracking performance, a modified barrier Lyapunov function based on a time-varying finite-time behavior boundary is constructed in the controller design, which can guarantee that the tracking error converges to a predetermined region within a specified time. Then by introducing the Nussbaum gain technique to handle the unknown control direction, an event-triggered neural output feedback control strategy is synthesized within the framework of dynamic surface control. Meanwhile, with the aid of Lyapunov synthesis, all the signals involved in the closed-loop system are proved to be bounded while Zeno phenomena is circumvented, and system outputs are well within the predefined region. Finally, an application on control design for a micro-electro-mechanical system gyroscope is given to validate the efficiency and superiority of proposed intelligent control scheme.

Stability and Stabilization of Takagi–Sugeno Fuzzy Systems With Hybrid Time-Varying Delays

This paper investigates the stability and stabilization of Takagi–Sugeno (T–S) fuzzy systems with discrete and distributed time-varying delays. First, $p$ th moment global exponential stability ( $p\geq 1$ ) of the addressed delayed fuzzy systems is considered by virtue of the comparison approach and inequality techniques. The developed algebraic criteria include some existing outcomes as special cases. Second, global exponential stabilization of the underlying delayed fuzzy systems is performed under a fuzzy state feedback controller. Third, considering that only a few studies have been concerned with finite-time stabilization of T–S fuzzy systems, by employing the comparison strategy and a nonlinear controller, finite-time stabilization of the nominated delayed fuzzy systems is presented. The result obtained herein establishes a general theoretical framework to analyze the finite-time behavior of delayed T–S fuzzy systems. Finally, simulation examples are conducted to illustrate the validity of the results.