Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Nondifferential Constraints

作     者：Miele, A. Damoulakis, J. N. Cloutier, J. R. Tietze, J. L. 

作者机构：Rice Univ Houston TX 77251 USA 

出 版 物：《JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS》 (优选法理论与应用杂志)

年 卷 期：1974年第13卷第2期

页      面：218-255页

核心收录：

学科分类：1201[管理学-管理科学与工程(可授管理学、工学学位)] 07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Office of Scientific Research  Office of Aerospace Research  United States Air Force [AF-AFOSR-72-2185] 

主　　题：Calculus of variations optimal control computing methods numerical methods gradient methods sequential gradient-restoration algorithm restoration algorithm boundary-value problems bounded control problems bounded state problems nondifferential constraints 

摘      要：This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the state. x(t), the control u(t), and the parameter pi so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, While the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), pi obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is de: ermined by a one-dimensional search on the augmented functional J, and the stepsize of the restoration phase by a one-dimensional search on the constraint error P. If alpha(g) is the gradient stepsize and alpha(r) is the restoration stepsize, the gradient corrections are of O(alpha(x)) and the restoration corrections are of O(alpha(r)alpha(2)(g)). Therefore. for alpha(g) sufficiently small the restoration phase preserves the descent property of the gradient phase: the functional (I) over cap at the end of any complete gradient-restoration cycle is smaller than the functional I at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time theta is replaced by the normalized time t, defined in such a way that the extremal arc has a normalized time length Delta t