The conjugate gradient method for optimal control problems with bounded control variables

Absfract-This paper extends the conjugate gradient minimization method of Fletcher and Reeves to optimal control problems. The technique is directly applicable only to unconstrained problems; if terminal conditions and inequality constraints are present, the problem must be converted to an unconstrained form; e.g., by penalty functions. Only the gradient trajectory, its norm, and one additional trajectory, the actual direction of search, need be stored. These search directions are generated from past and present values of the objective and its gradient. Successive points are determined by linear minimization down these directions, which are always directions of descent. Thus, the method tends to converge, even from poor approximations to the minimum. Since, near its m i n i m u m , a general nonlinear problem can be approximated by one with a linear system and quadratic objective, the rate of convergence is studied by considering this case. Here, the directions of search are conjugate and hence the objective is minimized over an expanding sequence of sets. Also, the distance from the current point to the miminum is reduced at each step. Three examples are presented to compare the method with the method of steepest descent. Convergence of the proposed method is much more rapid in all cases. A comparison with a second variational technique is also given in Example 3. T HIS PAPER presents an iterative procedure for solving unconstrained optimal control problems. Of course, a general formulation of the optimal control problem involves both terminal constraints on the state variables and inequalit!. constraints on the state and control variables enforced along the entire trajectory. Penalty functions have often been used to convert such problems to a sequence of " unconstrained " problems, i.e., problems with no terminal or inequality constraints [1]-[4]. I t is evident that the efficiencl-of these methods depends greatll-on the technique used to solve the unconstrained optimal control problem. Presently available techniques all have shortcomings. The convergence of steepest descent methods is often slon-[l ] n-hereas second-variational and Sen-ton methods may not converge at all. Thus there is strong motivation for developing more efficient means for solving unconstrained optimal control problems. Similar difficulties existed, until recent1)-, in the field of finite dimensional optimization, Le., mathematical programming. Hou-ever, in the past fen-years several rapid1)-convergent finite dimensional unconstrained minimization techniques have been developed. Among these are the method of Fletcher and Pori-ell [ 5 ] and the Fletcher-Reeves [6] …