Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques

A general system of the time-dependent partial differential equations containing several arbitrary initial and boundary conditions is considered. A hybrid method based on artificial neural networks, minimization techniques and collocation methods is proposed to determine a related approximate solution in a closed analytical form. The optimal values for the corresponding adjustable parameters are calculated. An accurate approximate solution is obtained, that works well for interior and exterior points of the original domain. Numerical efficiency and accuracy of the hybrid method are investigated by two-test problems including an initial value and a boundary value problem for the two-dimensional biharmonic equation.     

Infinite horizon linear quadratic Pareto game of the stochastic singular systems

This paper is concerned with the linear quadratic (LQ) Pareto game of the stochastic singular systems in infinite horizon. Firstly, the optimal control problem of the weighted sum cost functional is discussed. Utilizing the equivalent transformation method, the weighted sum LQ optimal control problem is transformed into a stochastic LQ optimization problem. Based on the classical stochastic LQ optimal control theory, the necessary and sufficient condition for the solvability of the indefinite weighted sum LQ optimal control is put forward. Then, the LQ Pareto game of the stochastic singular systems is studied. By the discussion of the convexity of the cost functionals, a sufficient condition for the existence of the Pareto solutions is obtained via the solvability of the corresponding generalized algebraic Riccati equation (GARE). Moreover, we derive all Pareto solutions based on the solution of a Lyapunov equation. Finally, an example is given to show the effectiveness of the proposed results.     

Properties of value function and existence of viscosity solution of HJB equation for stochastic boundary control problems

In the present paper, we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions, the continuity and differentiability of the value function are proved. We also define a new type of Hamilton–Jacobi–Bellman (HJB) equation and prove that the value function is a viscosity solution of this HJB equation.     

Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems

In this paper, we first develop an adaptive shifted Legendre–Gauss (ShLG) pseudospectral method for solving constrained linear time-delay optimal control problems. The delays in the problems are on the state and/or on the control input. By dividing the domain of the problem into a uniform mesh based on the delay terms, the constrained linear time-delay optimal control problem is reduced to a quadratic programming problem. Next, we extend the application of the adaptive ShLG pseudospectral method to nonlinear problems through quasilinearization. Using this scheme, the constrained nonlinear time-delay optimal control problem is replaced with a sequence of constrained linear-quadratic sub-problems whose solutions converge to the solution of the original nonlinear problem. The method is called the iterative-adaptive ShLG pseudospectral method. One of the most important advantages of the proposed method lies in the case with which nonsmooth optimal controls can be computed when inequality constraints and terminal constraints on the state vector are imposed. Moreover, a comparison is made with optimal solutions obtained analytically and/or other numerical methods in the literature to demonstrate the applicability and accuracy of the proposed methods.     

Stabilizing a class of mixed states for stochastic quantum systems via switching control

The global stabilization of a class of mixed states for finite dimensional stochastic quantum systems with degenerate measurement operator is investigated in this paper. We construct a measurement operator and control Hamiltonian that make the target state one of the system equilibria. Based on the proposed Lyapunov function, a control law is designed following Lyapunov’s method to steer system state to the target mixed state from an initial state in the convergence domain, which is obtained through the analyses of invariant set based on LaSella’s invariance principle. When the initial state isn’t in the convergence domain, a constant control is used to steer the system state so that it enters the convergence domain in finite time. The constant control and the control designed by Lyapunov method compose a switching control strategy, which can steer system state to the target mixed state from any arbitrary state in the state space, i.e., the target mixed state is globally stable under the switching control. The convergence of the switching control is proved based on state sample trajectories. Moreover, the numerical experiments on a three dimensional stochastic quantum system are performed to demonstrate the effectiveness of switching control.     

Infinite horizon multiobjective optimal control of stochastic cooperative linear-quadratic dynamic difference games

This article is concerned with the infinite horizon stochastic cooperative linear-quadratic (LQ) dynamic difference game in both the regular and the indefinite cases. Firstly, due to the constraints imposed on the weighting matrices and the linearity of the dynamic system, the costs are shown to be convex spontaneously for the regular stochastic cooperative LQ difference game, which yields the equivalence between the minimization of the weighted sum of costs and the Pareto optimal control. Secondly, the Pareto optimal control is derived for the regular game on the ground of the solution to the weighted algebraic Riccati equation (WARE) under exact observability, and then Pareto solutions are identified via the optimal feedback gain matrices and the solution to the weighted algebraic Lyapunov equation (WALE). Moreover, a new criterion which is also necessary and sufficient is developed to guarantee the costs to be convex for the indefinite case, and the Pareto optimality is investigated based on the solutions to the weighted generalized algebraic Riccati equation (WGARE) and the weighted generalized algebraic Lyapunov equation (WGALE) combining with the semidefinite programming (SDP). Finally, the fishery management game in the economy is presented to illustrate the obtained results.     

On an approximate effective speed of propagation

An algorithm is suggested for the determination of an effective speed of propagation for a random medium whose mean speed of propagation is perturbed by a small random term of the white-noise type. We study the propagation of disturbances in the limiting geometrical optics case, and pursue the analysis within the framework of stochastic optimal control. The statement of an appropriate Fermat's principle for the random medium and invoking the principle of dynamic programming lead to a non-linear elliptic equation—the classical Eikonal equation—perturbed by terms associated with the stochastics. This equation and estimates for the departure of the stochastic trajectories from the free-space geodesic path of propagation enable us to calculate the approximate speed.     

Unified Krylov-Bogoliubov-Mitropolskii method under a critical condition

A modified and compact form of Krylov-Bogoliubov-Mitropolskii (KBM) unified method is extended to obtain approximate solution of an nth order, n=2,3,…, ordinary differential equation with small nonlinearities when unperturbed equation has some repeated real eigenvalues. The existing unified method is used when the eigenvalues are distinct whether they are purely imaginary or complex or real. The new form is presented generalizing all the previous formulae derived individually for second-, third- and fourth-order equations to obtain undamped, damped, over-damped and critically damped solutions. Therefore, all types of oscillatory and non-oscillatory solutions are determined by suitable substitution of the eigenvalues in a general result. The formulation of the method is very simple and the determination of the solution is easy. The method is illustrated by an example of a fourth-order equation when unperturbed equation has two real and equal eigenvalues. The solution agrees with a numerical solution nicely. Moreover, this solution is useful when the differences between conjugate eigenvalues (real or complex) are small. Thus the method is a complement of the existing modified and compact form of KBM method.     

Noise suppresses exponential growth for neutral stochastic differential delay equations

Our aim in this paper is to investigate the polynomial growth for a class of neutral stochastic differential delay systems. We reveal that environmental noise will suppress the exponential growth if it is sufficiently strong. That is, the given system (neutral ordinary differential delay equation) whose solution grows exponentially and becomes a new system (neutral stochastic differential delay equation) whose solution will grow polynomially with probability one. We also provide two examples to illustrate the main result.     

Stochastic suppression and stabilization of nonlinear differential systems with general decay rate

In this paper, we investigate stochastic suppression and stabilization for a class of non-autonomous differential systems. Given a deterministic non-autonomous differential system, we introduce two independent Brownian motions and perturb this system into a new stochastic differential system. By using Lyapunov analysis method and some stochastic techniques, we show that a polynomial Brownian noise may guarantee the existence of global solution of the perturbed system while another linear Brownian noise may stabilize this system with general decay rate. An application of stochastic stabilization of differential system in the modeling of population growth is indicated.     

Dynamic programming,Fermat's principle and stochastic Eikonal equations

An extension of Fermat's principle to the stochastic case is presented in order to treat ray propagation in random media. The concept of dynamic programming permits one to derive a sequence of stochastic eikonal equations from which a sequence of rays can be traced using Hamilton's equations. The extension is motivated by analogous stochastic control problems in which the concepts and methods are adapted to the present problem. To facilitate understanding of the main ideas the presentation is given in a simple and somewhat naive manner. Two simple examples are presented to demonstrate the ideas and techniques.     

Linear quadratic open-loop Stackelberg game for stochastic systems with Poisson jumps

This paper is concerned with the finite horizon linear quadratic (LQ) Stackelberg game for stochastic systems with Poisson jumps under the open-loop information structure. First, the follower solves a LQ stochastic optimal control problem with Poisson jumps. With the aid of an introduced generalized differential Riccati equation with Poisson jumps (GDREP), the sufficient conditions for the optimization of the follower are put forward. Then, the leader faces an optimal control problem for a forward-backward stochastic differential equation with Poisson jumps (FBSDEP). By introducing new state and costate variables, a sufficient condition for the existence and uniqueness of the open-loop Stackelberg strategies is presented in terms of the solvability of two differential Riccati equations and a convexity condition. In addition, the state feedback representation of the open-loop Stackelberg strategies is obtained via the related differential Riccati equation. Finally, two examples shed light on the effectiveness of the obtained results.     

Stability of delayed Hopfield neural networks under a sublinear expectation framework

In this paper, we consider the stability of a class of stochastic delay Hopfield neural networks driven by G-Brownian motion. Under a sublinear expectation framework, we give the definition of exponential stability in mean square and construct some conditions such that the stochastic system is exponentially stable in mean square. Moreover, we also consider the stability of the Euler numerical solution of such equation. Finally, we give an example and its numerical simulation to illustrate our results.     

Iterated gain-based stochastic filters for dynamic system identification

We propose a novel form of nonlinear stochastic filtering based on an iterative evaluation of a Kalman-like gain matrix computed within a Monte Carlo scheme as suggested by the form of the parent equation of nonlinear filtering (Kushner–Stratonovich equation) and retains the simplicity of implementation of an ensemble Kalman filter (EnKF). The numerical results, presently obtained via EnKF-like simulations with or without a reduced-rank unscented transformation, clearly indicate remarkably superior filter convergence and accuracy vis-à-vis most available filtering schemes and eminent applicability of the methods to higher dimensional dynamic system identification problems of engineering interest.     

Adaptive finite-time control for stochastic nonlinear systems subject to unknown covariance noise

This paper is devoted to the adaptive finite-time control for a class of stochastic nonlinear systems driven by the noise of covariance. The traditional growth conditions assumed on the drift and diffusion terms are removed through a technical lemma, and the negative effect generated by unknown covariance noise is compensated by combining adaptive control technique with backstepping recursive design. Then, without imposing any growth assumptions, a smooth adaptive state-feedback controller is skillfully designed and analyzed with the help of the adding a power integrator method and stochastic backstepping technique. Distinctive from the global stability in probability or asymptotic stability in probability obtained in related work, the proposed design algorithm can guarantee the solution of the closed-loop system to be finite-time stable in probability. Finally, a stochastic simple pendulum system is skillfully constructed to demonstrate the effectiveness of the proposed control scheme.     

Mixed H∞ and passivity control for a class of stochastic nonlinear sampled-data systems

In this paper, we consider the problem of mixed H_∞ and passivity control for a class of stochastic nonlinear systems with aperiodic sampling. The system states are unavailable and the measurement is corrupted by noise. We introduce an impulsive observer-based controller, which makes the closed-loop system a stochastic hybrid system that consists of a stochastic nonlinear system and a stochastic impulsive differential system. A time-varying Lyapunov function approach is presented to determine the asymptotic stability of the corresponding closed-loop system in mean-square sense, and simultaneously guarantee a prescribed mixed H_∞ and passivity performance. Further, by using matrix transformation techniques, we show that the desired controller parameters can be obtained by solving a convex optimization problem involving linear matrix inequalities (LMIs). Finally, the effectiveness and applicability of the proposed method in practical systems are demonstrated by the simulation studies of a Chua’s circuit and a single-link flexible joint robot.     

一种基于完全线性变换法则的差分方程解析解算法

利用传统方法很难在计算机上实现差分方程的解析解求解,本文提出了一种获得差分方程解析解的线性算法,该算法的基础是完全线形变化法。其核心操作为降维处理,对高阶差分方程进行逐次降阶运算,直至获得其解析解表达式。本质上,该算法属于Z变换法的一种矩阵法变形。算法的线性特征使得其容易移植到计算机上实现差分方程的解析解运算,而非传统的数值迭代解。     

Approximations to the solution of linear Fredholm integrodifferential-difference equation of high order

There are few techniques available to numerically solve linear Fredholm integrodifferential-difference equation of high-order. In this paper we show that the Taylor matrix method is a very effective tool in numerically solving such problems. This method transforms the equation and the given conditions into the matrix equations. By merging these results, a new matrix equation which corresponds to a system of linear algebraic equation is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method. Moreover, this method is valid for the differential, difference, differential-difference and Fredholm integral equations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method.     

Analyzing noise-induced tracking errors in control systems with saturation: A stochastic linearization approach

Noise Induced Tracking Error (NITE) refers to the tracking error of the mean of the output in feedback control systems with nonlinear instrumentation subject to zero-mean measurement noise. Most of the previous work rely on the stochastic averaging for NITE analysis, the validity of which requires that the bandwidth of the zero mean measurement noise is much higher than that of the system. This is because the results obtained by stochastic averaging are asymptotic with respect to the noise bandwidth. Due to the asymptotic nature of the analysis tool, it is not straightforward to provide a quantitative argument for high bandwidth. An alternative method in the literature that can analyze NITE is stochastic linearization for random input, which is analogous to the well known describing function approach for sinusoidal input. Unlike stochastic averaging, stochastic linearization is not an asymptotic approximation. Therefore, analysis can be carried out for any given noise bandwidth. We carry out NITE analysis using stochastic linearization for a class of LPNI systems that are prone to NITE; identify the system conditions under which the averaging analysis of NITE may yield inaccurate results for a finite noise bandwidth; and prove that the results from the two methods agree as the noise bandwidth approaches infinity. In addition, an existing NITE mitigation strategy is extended based on the proposed method. Numerical examples are given to illustrate the results.     

Local exponential stabilization via boundary feedback controllers for a class of unstable semi-linear parabolic distributed parameter processes

This paper addresses the problem of local exponential stabilization via boundary feedback controllers for a class of nonlinear distributed parameter processes described by a scalar semi-linear parabolic partial differential equation (PDE). Both the domain-averaged measurement form and the boundary measurement form are considered. For the boundary measurement form, the collocated boundary measurement case and the non-collocated boundary measurement case are studied, respectively. For both domain-averaged measurement case and collocated boundary measurement case, a static output feedback controller is constructed. An observer-based output feedback controller is constructed for the non-collocated boundary measurement case. It is shown by the contraction semigroup theory and the Lyapunov’s direct method that the resulting closed-loop system has a unique classical solution and is locally exponentially stable under sufficient conditions given in term of linear matrix inequalities (LMIs). The estimation of domain of attraction is also discussed for the resulting closed-loop system in this paper. Finally, the effectiveness of the proposed control methods is illustrated by a numerical example.