Modified quasilinearization and optimal initial choice of the multipliers part 2—Optimal control problems

This paper considers the problem of extremizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . The state is ann-vector, the control is anm-vector, and the parameter is ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. A modified quasilinearization algorithm is developed; its main property is a descent property in the performance indexR, the cumulative error in the constraints and the optimum conditions.Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of a scaling factor (or stepsize) in the system of variations. The stepsize is determined by a one-dimensional search so as to ensure the decrease in the performance indexR; this can be achieved through a bisection process starting from = 1. Convergence is achieved whenR becomes smaller than some preselected value.In order to start the algorithm, some nominal functionsx(t),u(t), and multipliers (t), must be chosen. In a real problem, the selection ofx(t),u(t), can be made on the basis of physical considerations. Concerning (t) and , no useful guidelines have been available thus far. In this paper, a method for selecting (t) and optimally is presented: the performance indexR is minimized with respect to (t) and . Since the functionalR is quadratically dependent on (t) and , the resulting variational problem is governed by Euler equations and boundary conditions which are linear.Two numerical examples are presented, and it is shown that, if the initial multipliers (t) and are chosen optimally, modified quasilinearization converges rapidly to the solution. On the other hand, if the initial multipliers are chosen arbitrarily, modified quasilinearization may or may not converge to the solution. From the examples, it is concluded that the beneficial effects associated with the optimal initial choice of the multipliers (t) and lie primarily in increasing the likelihood of convergence rather than accelerating convergence. However, this optimal choice does not guarantee convergence, since convergence depends on the functional being extremized, the differential constraints, the boundary conditions, and the nominal functionsx(t),u(t), chosen in order to start the algorithm.This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1. The authors are indebted to Mr. E. E. Cragg for analytical and computational assistance.