Sequential gradient-restoration algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter 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 functionsx(t),u(t), 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.Here, the state inequality constraint is handled in a direct manner. A predetermined number and sequence of subarcs is assumed and, for the time interval for which the trajectory of the system lies on the state boundary, the control is determined so that the state boundary is satisfied. The state boundary and the entrance conditions are assumed to be linear inx and , and the sequential gradient-restoration algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration of the gradient phase and the restoration phase along all of the subarcs composing the trajectory.At first glance, the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory. Actually, this is not the case. The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variables.To facilitate the numerical solution on digital computers, the actual time is replaced by the normalized timet, defined in such a way that each of the subarcs composing the extremal arc has a normalized time length t=1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times ₁, ₂, at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal boundary is reached are regarded to be components of the parameter being optimized.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.This paper is based in part on a portion of the dissertation which the first author submitted in partial fulfillment of the requirements for the PhD Degree at the Air Force Institute of Technology, Wright-Patterson AFB, Ohio. This research was supported in part by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Professor H. Y. Huang, Dr. R. R. Iyer, Dr. J. N. Damoulakis, Mr. A. Esterle, and Mr. J. R. Cloutier for helpful discussions as well as analytical and numerical assistance. This paper is a condensation of the investigations reported in Refs. 1–2.