Pontryagin Equation Research Articles

Stochastic Analysis for the Embedded Single-Walled Carbon Nanotube Under Random Vibrations

It is inevitable that carbon nanotubes (CNTs) are subjected to random vibrations due to environmental noise, which may impair their mechanical properties. The nonlinear dynamics associated with single-walled carbon nanotubes (SWCNTs) under random noise has been investigated in this work. First, based on the nonlocal theory and the Euler–Bernoulli beam model with fixed supports at both ends, the governed equations and boundary conditions are established according to Hamilton’s principle. Second, the Galerkin method is used to approximate the differential equations of motion and the stochastic averaging method (SAM) is applied to predict the approximate response of the original system. Subsequently, a detailed parametric study is conducted to analyze the nonlinear random vibration response of the SWCNT with nonlocal effects, and the effectiveness of the proposed method is verified by comparing the analytical results with those from the Runge–Kutta method. Finally, by solving the backward Kolmogorov (BK) equation and the generalized Pontryagin (GP) equation simultaneously, the conditional reliability function (CRF) and mean first-passage time (MFPT) for determining the reliability of SWCNT systems are obtained.

On non-conservative perturbations of three-dimensional integrable systems

At present, non-conservative perturbations of two-dimensional nonlinear Hamiltonian systems have been studied quite fully. The purpose of the study is to generalize this theory to the three-dimensional case, when the unperturbed system is nonlinear, integrable and has a region filled with closed phase trajectories. In this paper, autonomous perturbations are considered and the main attention is paid to the problem of limit cycles. Methods. The study is based on the construction of special coordinates in which the variables are divided into two slow and one fast, and in the first approximation with respect to a small parameter the equations for the slow variables are separated. Results. It is shown that hyperbolic equilibrium states of a truncated system determine closed phase trajectories, in the vicinity of which cycles appear under the perturbation. Conclusion. Thus, the problem is reduced to the study of solutions of the “generating” system of two algebraic or transcendental equations, similar to the generating Poincare–Pontryagin equation for two-dimensional systems. As examples, we considere a three-imensional van der Pol type system and the Lorentz system in the case of large Rayleigh numbers.

Reliability of quasi-integrable Hamiltonian systems with coupled viscoelastic dampings driven by wideband noises

Viscoelasticity is the time-dependent anelastic behavior of materials, which means the response of a viscoelastic system to a stimulus depends on the instantaneous state and its history. For quasi-integrable Hamiltonian systems with coupled viscoelastic dampings driven by wideband noises, the key to estimate their reliability is how to handle the coupled viscoelastic dampings between different degrees of freedom. In the present paper, the contributions of the coupled viscoelastic dampings are decoupled into elastic restoring forces and viscous dampings by using the generalized harmonic balance technique, and the equivalent quasi-integrable Hamiltonian systems without coupled viscoelastic dampings are obtained. Then by using the stochastic averaging method, the averaged equations for all slowly varying processes of the equivalent systems in both non-resonant and internally resonant cases are derived, which retain the main dynamical characteristics of the equivalent systems. The lower-dimensional backward Kolmogorov (BK) equation for reliability and the Pontryagin equation for the mean first passage time of the equivalent systems are derived. Two examples are worked out in detail to illustrate the application of the proposed procedure. Comparisons of the results obtained by using the proposed method with those from the Monte Carlo simulations of original systems show that the proposed method is well applicable.

Optimal Finite-Dimensional Controller of the Stochastic Differential Object’s State by Its Output. I. Incomplete Precise Measurements

The well-known problem of synthesizing the optimal on the average and on given time interval of the inertial control law for a continuous stochastic object if only a part of its state variables are accurately measured is considered. Due to the practical unrealizability of its classical infinite-dimensional Stratonovich-Mortensen solution, it is proposed to limit ourselves to optimizing the structure of a finite-dimensional dynamic controller, whose order is chosen by the user. This finiteness allows using a truncated version of the a posteriori probability density that satisfies a deterministic partial differential integrodifferential equation. Using the Krotov extension principle, sufficient optimality conditions for the structural functions of the controller and the Lagrange–Pontryagin equation for finding their extremals are obtained. It is shown that in particular cases of the absence of measurements, complete measurements and taking into account only the values of incomplete measurements, the proposed controller turns out to be static (inertialess), and the relations for its synthesis coincide with the known ones. For a dynamic controller, algorithms for finding each of its structural functions are given.

Stochastic Dynamics of Suspension System in Maglev Train: Governing Equations for Response Statistics and Reliability

The suspension system of the maglev train will inevitably be disturbed by random factors such as track irregularities, which will cause random vibration of the train and even affect the safety of the train. Therefore, the research on the response and reliability of suspension system under random disturbance is crucial to its safe operation. In this paper, the response and the reliability of a suspension system are investigated using the theory and methods of stochastic dynamics. First, the magnetic gap and vertical velocity of the suspension system are random due to the random disturbance. Thus, the stochastic response is investigated through the probability density function (PDF), which is governed by the Fokker–Planck–Kolmogorov (FPK) equation corresponding to suspension system. And the response statistics of the suspension system under different system parameters and disturbance intensities are analyzed by solving the corresponding FPK equation using the finite difference (FD) method. Second, random disturbance may lead to the vibration amplitude of the suspension system exceeding the safety domain and causing safety incident, which is a reliability problem in stochastic dynamical systems. The probability that response is still in the safety domain at a given time is the reliability function of the suspension system, which is governed by the backward Kolmogorov equation. The time that the response first passes through the safety domain is the first-passage time, and its n-order moment satisfies the generalized Pontryagin equation. Reliability of the suspension system is analyzed by solving these governing equations using the FD method. In addition, the results of the FD method in this paper are verified with those of Monte Carlo (MC) simulation, which shows the correctness of FD method.

Reliability of the bi-stable piezoelectric energy harvester under random excitation

The bi-stable piezoelectric energy harvester can produce higher voltage and efficiency than traditional energy harvester. In this manuscript, the reliability of the bi-stable piezoelectric energy harvester excited by Gaussian white noise is investigated. The equivalent uncoupled equation of system is deduced on the basis of the generalized harmonic transformation for the bi-stable piezoelectric energy harvester original coupling system. The averaged stochastic differential equation for system energy of the equivalent uncoupled system is established using the stochastic averaging method. The conditional reliability function and the mean first-passage time are derived through the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged stochastic differential equation. The influences of noise intensity, electromechanical coupling coefficient, and time constant ratio on the conditional reliability function and mean first-passage time are discussed. The analytical results are highly consistent with the numerical results that are obtained by the Monte Carlo numerical simulation.

Optimal control of compartmental models: The exact solution

We formulate a control problem for positive compartmental systems formed by nodes (buffers) and arcs (flows). Our main result is that, on a finite horizon, we can solve the Pontryagin equations in one shot without resorting to trial and error via shooting. As expected, the solution is bang–bang and the switching times can be easily determined. We are also able to find a cost-to-go-function, in an analytic form, by solving a simple nonlinear differential equation. On an infinite horizon, we consider the Hamilton–Jacobi–Bellman theory and we show that the HJB equation can be solved exactly. Moreover, we show that the optimal solution is constant and the cost-to-go function is linear and copositive. This function is the solution of a nonlinear equation. We propose an iterative scheme for solving this equation, which converges in finite time. We also show that an exact solution can be found if there is a positive external disturbance affecting the process and the problem is formulated in a minsup framework. We finally provide illustrative examples related to flood control and epidemiology.

Reliability analysis of a class of stochastically excited nonlinear Markovian jump systems

It is inevitable for the nonlinear systems to suffer from external stochastic disturbance. Meanwhile, the components failure will bring abrupt changes in its substructures, which can be considered as the internal stochastic disturbance. It is demonstrated that the components failure performs random jumpy factors switching between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of nonlinear hybrid systems undergoing Markovian jumps. In this paper a novel method is presented to analyze the dynamical reliability of multi-degree-of-freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. Firstly, the Markovian jump process is introduced to formulate the aforementioned systems as continuous-discrete hybrid systems. Secondly, a two-step approximate method is applied to convert the hybrid systems to one governed by a set of averaged Itô stochastic differential equations (SDEs) of mechanical energies. Then the associated averaged backward Kolmogorov equation and the Pontryagin equation are constructed and solved to yield the conditional reliability function and the mean first-passage time of the original system respectively. Finally, an example of two coupled Duffing oscillators is illustrated to compare the analytical results and those from Monte Carlo simulation to verify the effectiveness of the proposed method.

The analysis of the dynamic optimization problem in econophysics from the point of view of the symplectic approach for constrained systems

A standard approach to deal with the problems relative to dynamic optimization is the well known Pontryagin method to obtain a concise condition for an optimal control, minimizing the cost functional. In this paper we have proposed a simpler dynamic optimization procedure through a so-called symplectic algorithm. We worked with the analogy between the physical systems at the classical and quantum energy levels. In this way, we started by considering some cost functional f(q,u,t), where q and u are state and control variables, respectively. We have shown that it is possible to investigate the system by means of a symplectic extension, where we can reduce any constrained system into its canonical first order form. Consequently, the dynamics of evolution, usually governed by Dirac’s constraint method, was re-obtained here in a very elegant way. On the other hand, the constraints classification turned out to be different from the one used in Dirac’s procedure. Of course, our analysis is valid for unconstrained systems where the Pontryagin equations are valid too. Four optimization problems were solved.

Reliability and control of strongly nonlinear vibro-impact system under external and parametric Gaussian noises

This paper mainly focuses on reliability and the optimal bounded control for maximizing system reliability of a strongly nonlinear vibro-impact system. Firstly, the new stochastic averaging in which the impact condition is converted to the system energy is applied to obtain the averaged Ito stochastic differential equation, by which the associated Backward Kolmogorov (BK) equation and Generalized Pontryagin (GP) equation are derived. Then, the dynamical programming equations are obtained based on the dynamical programming principle, by which the optimal bounded control for maximizing system reliability is devised Finally, the effects of the bounded control and noise intensity on the reliability of the vibro-impact system are discussed in detail; meanwhile, the influence of impact conditions on the system’s reliability is also studied. The feasibility and effectiveness of the proposed analytical method are substantiated by numerical results obtained from Monte-Carlo simulation.

Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation

The reliability of quasi integrable and non-resonant Hamiltonian system under fractional Gaussian noise (fGn) excitation is studied. Noting rather flat fGn power spectral density (PSD) in most part of frequency band, the fGn is innovatively regarded as a wide-band process. Then, the stochastic averaging method for quasi integrable Hamiltonian systems under wide-band noise excitation is applied to reduce 2n-dimensional original system into n-dimensional averaged Ito stochastic differential equations (SDEs). Reliability function and mean first passage time are obtained by solving the associated backward Kolmogorov equation and Pontryagin equation. The validity of the proposed procedure is tested by applying it to an example and comparing the numerical results with those from Monte Carlo simulation.

Reliability of elastic impact system with Coulomb friction excited by Gaussian white noise

Abstract The reliability of nonlinear Elastic impact oscillator with Coulomb friction subjected to Gaussian white noise excitation is investigated. Firstly, the stochastic averaging method for the nonlinear Elastic impact system with Coulomb friction is proposed and the averaged Ito equation for the system energy is obtained. Secondly, the Backward Kolmogorov(BK) equation and Generalized Pontryagin(GP) equation are established to achieve the conditional reliability function and the mean first-passage time of the Elastic impact system. Finally, two examples are worked out in detail to illustrate the feasibility of the proposed procedures. The accuracy and efficiency of the proposed procedures are validated by Monte Carlo simulations. We can conclude that the increase of the Coulomb friction, noise intensity and damping constants of the elastic impact force can improve the reliability of the Elastic impact system.

A New Discrete Analogue of Pontryagin’s Maximum Principle

By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation.

The quantum dark side of the optimal control theory

In a recent article, a generic optimal control problem was studied from a physicist’s point of view (Contreras et al. 2017). Through this optic, the Pontryagin equations are equivalent to the Hamilton equations of a classical constrained system. By quantizing this constrained system, using the right ordering of the operators, the corresponding quantum dynamics given by the Schrödinger equation is equivalent to that given by the Hamilton–Jacobi–Bellman equation of Bellman’s theory. The conclusion drawn there were based on certain analogies between the equations of motion of both theories.In this paper, a closer and more detailed examination of the quantization problem is carried out, by considering three possible quantization procedures: right quantization, left quantization, and Feynman’s path integral approach. The Bellman theory turns out to be the classical limit ħ→0 of these three different quantum theories. Also, the exact relation of the phase S(x,t) of the wave function Ψ(x,t)=eiħS(x,t) of the quantum theory with Bellman’s cost function J+(x,t) is obtained. In fact, S(x,t) satisfies a ‘conjugate’ form of the Hamilton–Jacobi–Bellman equation, which implies that the cost functional J+(x,t) must necessarily satisfy the usual Hamilton–Jacobi–Bellman equation. Thus, the Bellman theory effectively corresponds to a quantum view of the optimal control problem.

Dynamical reliability of internally resonant or non-resonant strongly nonlinear system under random excitations

Abstract The dynamical reliability of multi-degrees-of-freedom (MDOF) strongly nonlinear system under Gaussian white noise excitations is studied, including resonance and non-resonance. Firstly, the equations of motion of the original system with or without internal resonance are reduced to a set of Ito stochastic differential equations after stochastic averaging. Then, the backward Kolmogorov equation and the Pontryagin equation associated with the resonantly or non-resonantly averaged Ito stochastic differential equations, which determine the conditional reliability function and the mean first-passage time of the original random system, are constructed under appropriate boundary and (or) initial conditions, respectively. In particular, if the non-resonantly averaged system is completely decoupled, the conditional reliability function and the mean first-passage time of the original non-resonant system can be obtained by solving a set of simplified backward Kolmogorov equations. A system comprising two weakly coupled and strongly nonlinear mechanical oscillators is given as a concrete example to show the application of the proposed method. The 1:1 internal resonance or non-resonance is discussed. The corresponding high-dimensional backward Kolmogorov equation and Pontryagin equation are established and solved numerically. All theoretical results are validated by a Monte Carlo digital simulation.

Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.

First-Crossing Problem of Weakly Coupled Strongly Nonlinear Oscillators Subject to a Weak Harmonic Excitation and Gaussian White Noises

We study first-crossing problem of two-degrees-of-freedom (2DOF) strongly nonlinear mechanical oscillators analytically. The excitation is the combination of a deterministic harmonic function and Gaussian white noises (GWNs). The generalized harmonic function is used to approximate the solutions of the original equations. Four cases are studied in terms of the types of resonance (internal or external or both). For each case, the method of stochastic averaging is used and the stochastically averaged Itô equations are obtained. A backward Kolmogorov (BK) equation is set up to yield the failure probability and a Pontryagin equation is set up to yield average first-crossing time (AFCT). A 2DOF Duffing-van der Pol oscillator is chosen as an illustrative example to demonstrate the effectiveness of the analytical method. Numerically analytical solutions are obtained and validated by digital simulation. It is shown that the proposed method has high efficiency while still maintaining satisfactory accuracy.

First-passage failure of linear oscillator with non-classical inelastic impact

• Stochastic averaging and first-passage failure of vibro-impact system are investigated. • The bigger parameters ζ, s p and k 2 can enhance the system reliability. • The uncorrelated random noises seem more easily destroy the system reliability. The first-passage failure of linear oscillator with inelastic impact subjected to the additive and multiplicative random noises is investigated. The impact is described by the non-classical inelastic impact model, which is essentially different from the traditional impact model and can provide the whole information of the impact process. First of all, the impact force in the motion equation is replaced by the quasi-linear damping and nonlinear stiffness terms. Then, the stochastic averaging is adopted and the averaged Itô stochastic deferential equation of the total system energy is derived. Last, by solving the established backward Kolmogorov equation and Pontryagin equation from the averaged Itô equation numerically, the conditional reliability, the conditional probability density function (PDF) and the mean time of first-passage failure can be obtained. The comparison between the analytical results and those from Monte-Carlo simulation reveals the proposed procedure is effective. The influences of some system parameters are discussed in detail.

Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations

A linear oscillator simultaneously subjected to stochastic forcing and parametric excitation is considered. The time required for this system to evolve from a low initial energy level until a higher energy state for the first time is a random variable. Its expectation satisfies the Pontryagin equation of the problem, which is solved with the asymptotic expansion method developed by Khasminskii. This allowed deriving closed-form expressions for the expected first passage time. A comprehensive parameter analysis of these solutions is performed. Beside identifying the important dimensionless groups governing the problem, it also highlights three important regimes which are called incubation, multiplicative and additive because of their specific features. Those three regimes are discussed with the parameters of the problem.

Delay time for fine particle ignition within gas with fluctuating temperature

The Pontryagin equation was applied to calculating the average time for the random process escaping the assign interval: this gives the average delay time for waiting of particle ignition moment in a turbulent flow of gas. A direct numerical simulation method was developed for gas temperature fluctuations with assigned autocorrelation function and particle temperature fluctuations due to exothermal chemical reaction. The method was based on numerical solution of a system of stochastic differential equations. Results of direct simulation were validated through comparing with the analytical solution available for particles without exothermal reaction. Analytical calculations and results of direct numerical simulation for the delay time of particle ignition are in agreement.