Terminal Optimal Control Problem Research Articles

Эволюционные вычисления для решения задачи терминального оптимального управления

The present article considers the problem of numerical solution of the terminal optimal control problem. The general statement of the terminal optimal control problem and a brief overview of its solving methods are presented. With a direct approach and reduction of the optimal control problem to the finite-dimensional optimization problem, the target functional on the space of desired parameters, regardless of the type of approximation of the control function, may not have the unimodal property. Therefore, it is advisable to use evolutionary algorithms to solve the problem. A general approach to solving the terminal optimal control problem of evolutionary computational algorithms is presented. The paper presents a description of some evolutionary algorithms that were selected as the most effective for solving the optimal control problem. A hybrid evolutionary algorithm based on a combination of several evolutionary algorithms is considered. The computational experiment considers the terminal optimal control problems, for which optimal solutions were found by known classical numerical methods that use the gradient of the target functionality when searching. Comparison of the results obtained by classical and evolutionary methods by functional values and computational costs allows us to conclude that evolutionary algorithms are able to effectively solve the terminal optimal control problems

Numerical methods for solving terminal optimal control problems

Numerical methods for solving optimal control problems with equality constraints at the right end of the trajectory are discussed. Algorithms for optimal control search are proposed that are based on the multimethod technique for finding an approximate solution of prescribed accuracy that satisfies terminal conditions. High accuracy is achieved by applying a second-order method analogous to Newton’s method or Bellman’s quasilinearization method. In the solution of problems with direct control constraints, the variation of the control is computed using a finite-dimensional approximation of an auxiliary problem, which is solved by applying linear programming methods.

Terminal control of boundary models

A terminal optimal control problem for finite-dimensional static boundary models is formulated. The finite-dimensional models determine the initial and terminal states of the plant. The choice of an optimal control drives the plant from one state to another. A saddle-point method is proposed for solving this problem. The convergence of the method in a Hilbert space is proved.

A parametric convex optimal control problem for a linear system

The dependence of the solutions of a terminal optimal control problem on a parameter in the initial state vector is investigated. Attention is devoted mainly to the behaviour of the solution in the neighbourhood of a non-regular point. On the basis of the results, a method is proposed for constructing solutions of the problem for all parameter values. In practical work it is often important to know not only the solution of an optimal control problem for fixed parameter values, but also the dependence of the solution on the parameters, which enables one to estimate how the solution may vary when the parameters fluctuate. In addition, a knowledge of the dependence of the solutions of optimal control problems on the parameters provides the basis for methods of constructing feedback controls [1, 2], as well as stabilization and estimation methods based on the moving horizon strategy [3–5]. Numerical solutions of such problems are generally achieved by continuation of the solution with respect to a parameter [6–9]. The greatest difficulties in applying such methods arise in the case when the “actual” value of the parameter is a non-regular point. Therefore, in most publications devoted to sensitivity analysis and to investigating the parameter-dependence of the solutions, it is assumed that all parameter values are regular, or of degree of non-regularity one. In this paper no such assumptions are made.

Asymptotic solution of the optimal control problem for standard systems with delay

The authors consider the construction of an asymptotic solution of the terminal optimal control problem using the averaging method. The optimal process is described by the equation z = eZ (t, z, z(t-l, e, u), u), z/t=[-1,0] = {var_phi}(t), where the delay is constant and of unit magnitude, z {element_of} G is an n-dimensional vector, G {contained_in} R{sup n}, e > 0 is a small parameter, t {element_of} T {triple_bond} [0, e{sup -1}], Z {var_phi} are n-dimensional vector functions, Z is strictly convex in u for any (t, z) {element_of} T X G, u {element_of} U is the r-dimensional control vector, U is a compact set.

Consideration of Optimal Control Problems Specificity On Generalizing Mathematical Programming Algorithms

New ways of constructing algorithms for optimal control problems solution on the basis of special modifications of mathematical programming algorithms are investigated (R.Gabasov, F.M.Kirillova. Linear Programming Methods. Parts I-III, BGU Publishing House, Minsk, 1977, 1978, 1980). The paper develops the authors' results reported at the International IFAC Congresses (Helsinki, 1978; Kyoto, 1981) and at International IFAC Workshops and Symposia (Sochi, 1980; Denver, 1979; Madrid, 1982). Optimization algorithms in the class of impulse and piece-wise continuous controls where control problems are not reduced to finite-dimensional ones are considered. Dynamic systems, regularly and singularly perturbed systems, systems with rapid-changing parameters are the objects of optimization. Terminal optimal control problems, optimal problems with phase constraints, non-smooth optimal control problems, nonlinear optimal control problems are examined. Primal and dual algorithms are worked out. The results of computer experiments are presented.

Modifications of the method of successive approximations for solving optimal control problems

THE TERMINAL optimal control problem when the trajectory has a free right-hand end is considered. A new method is described for finding a set of small measure, in which the control is varied finitely so as to reduce the value of the functional. Estimates are obtained and convergence theorems are proved with certain assumptions.