On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls

The wide class of nonlinear stationary systems of ordinary differential equations taking into account restrictions on control and external perturbation is considered. An algorithm for constructing a discrete control function that guarantees the transfer of the systems from the initial state to the origin and an arbitrary neighborhood of the origin is proposed. A constructive sufficient condition of the Kalman type, in which the specified translation is possible, is obtained. The problem of robot‐manipulator control is considered and its numerical simulation is carried out.     

Trajectory optimization for maneuvering satellites

This paper proposes an algorithm to compute optimal trajectories for a maneuvering satellite by using a nonlinear programming representation of an optimal control problem. In this problem, a satellite must be located at a given final position with given velocity from initial position and velocity without passing a prohibited region such as the atmosphere while achieving minimum fuel consumption. Optimal control theory is applied to obtain a set of ordinary differential equations subject to two-point boundary conditions (TPB) on the adjoint system. Then an exact penalty function method is employed to obtain the optimal trajectories by solving the TPB problem as initial conditions for the adjoint system and an unknown final time are regarded as decision variables. This formulation, where the optimal control technique and the nonlinear programming method are incorporated, permits more systematic and flexible algorithm implementation.     

A method and a software package for numerical solution of the systems of nonlinear ordinary integro-differential-algebraic equations

A method, an algorithm and a software package for automatically solving the ordinary nonlinear integro-differential-algebraic equations (IDAEs) of a sufficiently general form are described. The author understands an automatic solution as obtaining a result without carrying out the stages of selecting a method, programming, and program checking. Both initial and boundary value problems for such equations are solved. It is assumed that the complete set of boundary and initial conditions at the beginning of the integration interval are given. By performing differentiation, the system of IDAEs can be modified, in general, into a system of ordinary nonlinear differential equations (IDEs). The problem of finding the solution of the above-mentioned system on the uniform grid on the integration interval is posed in two forms: as solving the system of IDAEs and as solving the appropriate system of IDEs, where the developed program is to be used. In order to reduce the system of IDAEs and the system of IDEs to the systems of ordinary nonlinear algebraic equations, at every stage of the algorithm the integration and differentiation formulas obtained earlier by N.G. Bandurin are used. Systems similar to those test systems of both nonlinear IDAEs and IDEs considered in this investigation are solved by using the computer programs. It is evident that the coincidence of the results for one and the same system of equations in its different forms can serve as good evidence of the correctness of the obtained results.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Artificial neural network approach for solving fuzzy differential equations

The current research attempts to offer a novel method for solving fuzzy differential equations with initial conditions based on the use of feed-forward neural networks. First, the fuzzy differential equation is replaced by a system of ordinary differential equations. A trial solution of this system is written as a sum of two parts. The first part satisfies the initial condition and contains no adjustable parameters. The second part involves a feed-forward neural network containing adjustable parameters (the weights). Hence by construction, the initial condition is satisfied and the network is trained to satisfy the differential equations. This method, in comparison with existing numerical methods, shows that the use of neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.     

Metaheuristic algorithms for approximate solution to ordinary differential equations of longitudinal fins having various profiles

Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. Approximate approaches have been utilized when obtaining analytical (exact) solutions requires substantial computational effort and often is not an attainable task. Hence, the importance of approximation methods, particularly, metaheuristic algorithms are understood. In this paper, a novel approach is suggested for solving engineering ordinary differential equations (ODEs). With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (error function) of the ODEs. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance and inverted generational distance metrics are used for evaluation and assessment of the approximate solutions versus the exact (numerical) solutions. Longitudinal fins having rectangular, trapezoidal, and concave parabolic profiles are considered as studied ODEs. The optimization task is carried out using three different optimizers, including the genetic algorithm, the particle swarm optimization, and the harmony search. The approximate solutions obtained are compared with the differential transformation method (DTM) and exact (numerical) solutions. The optimization results obtained show that the suggested approach can be successfully applied for approximate solving of engineering ODEs. Providing acceptable accuracy of the proposed technique is considered as its important advantage against other approximate methods and may be an alternative approach for approximate solving of ODEs.     

The FEM optimization of active vibration control in structures by solving the corresponding two-point-boundary-value problem

An algorithm for solving optimal active vibration control problems by the finite element method (FEM) is presented. The optimality equations for the problem are derived from Pontryagin’s principle in the form of a set of the fourth order ordinary differential equations that, together with the initial and final boundary conditions, constitute the boundary value problem in the time domain, which in control is referred to as a two-point-boundary-value problem. These equations decouple in the modal space and can be solved by the FEM technique. An analogy between the optimality equations and the governing equations for a set of certain static beams permits obtaining numerical solutions to the optimal control problem with the help of standard ‘structural’ FEM software. The optimal action of actuators is automatically calculated by applying the independent modal space control concept. The structure’s response to actuation forces is also determined and can independently be verified for spillover effects. As an illustration, the algorithm is used for the analysis of optimal action of actuators to attenuate vibrations of an elastic fin.     

The recursive reduced-order numerical solution of the singularlyperturbed matrix differential Riccati equation

Under stability-observability conditions imposed on a singularly perturbed system, an efficient numerical method for solving the corresponding matrix differential Riccati equation is obtained in terms of the reduced-order problems. The order reduction is achieved via the use of the Chang transformation applied to the Hamiltonian matrix of a singularly perturbed linear-quadratic control problem. An efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation     

一种解列车节能操纵问题的改进算法

本文提出一种解列车节能操纵问题的算法。该算法是基于微分方程的数值解和解带约束条件的优化问题的遗传算法。     

一种解列车节能操纵问题的改进算法

本文提出一种解列车节能操纵问题的算法。该处是基于微分方程的数值解和解带约束条件的优化问题的遗传算法。     

Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

Necessary optimality conditions in a nonsmooth control problem with variable structure

An optimal control problem with nonsmooth performance criterion described by a system of ordinary differential equations is considered. Necessary first-order optimality conditions are obtained.     

Numerical solving of control problem for object with uncertain information on its initial state and parameters

Problems of the optimal control of objects described by systems of ordinary differential equations in the class of piecewise constant control functions at inexact initial information about the values of the initial conditions and object parameters are investigated in this article. The piecewise constant values of the controls and the boundaries of the controls’ constancy intervals are optimized in this problem. The necessary optimality conditions and formulas for the functional gradient that allow using effective first-order optimization procedures for the numerical solution at a given number of the controls constancy intervals are obtained. The algorithm of its optimization is offered for the case when the number of intervals of constancy is not give.     

Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解

对于中厚板的静力弯曲和自由弯曲振动问题,引入两个辅助函数,采用胡海昌在Reissner板理论基础上提出的中厚板微分方程及边界条件,将两类问题的控制方程引入Hamilton体系,分别得到Hamilton体系下中厚板静力弯曲和自由振动问题的微分方程组模型. 比较后得到了Hamilton体系下中厚板静力和振动问题的统一模型,其特点是: 微分方程组模型的统一形式中Hamilton矩阵在对角线位置有2个零子块矩阵. 对于中厚板静力和振动问题,比较了所得齐次微分方程组的特征根,给出齐次微分方程组的通解并进行了比较,从而使问题的求解更理性化和合理化,求解过程遵循一套统一的方法论,便于把这类解法推广到其它问题.     

Numerical Solution to a Class of Inverse Problems for Parabolic Equation

A class of inverse problems for parabolic equation is considered. In particular, boundary value problems with nonlocal conditions are reduced to such class of problems. The proposed numerical approach is based on the method of lines to reduce the problem to a system of ordinary differential equations. To solve this system, an analog of the transfer method for boundary conditions is applied. The results of numerical experiments are given.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

A method of iteration for boundary value solution of ordinary differential equations

A globally stable iterative search algorithm for finding the unknown initial values in the two-point boundary-value problem (TPBVP) arising in ordinary differential equations has been developed. The problem can be embedded in a discrete-time feedback control system, the steady state of which is the required solution. Algorithms are found through the construction of globally stable digital controllers.     

A hybrid method for the solution of first order partial differential equations arising in non-linear estimation

A method for the hybrid computer solution of the Cauchy problem for a general first order partial differential equation is presented. The method is based on the well known theory of characteristics and the solution is obtained by generating the characteristic curves of the partial differential equation. The problem is thus transformed to that of integrating a system of ordinary differential equations using different initial conditions. The organization of the computation is such that the analog computer integrates the system of ordinary differential equations whereas the digital computer commands the operation, supplies initial conditions and stores the results. The method has been illustrated by simulating two partial differential equations arising in non-linear estimation.     

A model of optimal control over a nonlinear multidimensional innovation diffusion process

A model of competitive innovation diffusion is considered. The model is based on the Lotka-Volterra system and an initial-boundary problem for a system of quasilinear parabolic equations. The maximum principle is proved for the problem of diffusion of two competitive innovations, and sufficient conditions of existence of optimum control are obtained for the system. A numerical algorithm is constructed for solving optimum control problems, and numerical results for a model example are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 120–133, July–August 2008.