The computation of optimal controls in the presence of equality nondifferential constraints [by] James Robert Cloutier
Cloutier, J. R.
Master of Arts
An algorithm is developed to solve optimal control problems involving a functional I subject to differential constraints, terminal constraints, and equality nondifferential constraints. The algorithm is composed of a sequence of cycles, each cycle consisting of two phases, a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the value of the functional while satisfying the constraints to first order. During this iteration, the gradient is projected onto the tangential hyperplane of the constraints, and a step is taken in the negative direction of the projection. The restoration phase involves one or more iterations and is designed to restore the constraints to a predetermined accuracy while minimizing the norm of the variations of-the control and the parameter. To achieve this constraint satisfaction, quas-ilinearization (Newton's method) is employed. The gradient stepsize is chosen sufficiently small so that the restoration phase preserves the descent property of the gradient phase. This is possible due to the fact that the gradient corrections are of (a ) while the restoration corrections are of (a). Thus, the value of the functional f at the end of g any cycle is smaller than the value of the functional I at the beginning of the cycle. The restoration phase is terminated, or bypassed as the case may be, whenever the norm of the error in the constraints is less than its predetermined tolerance. Convergence is attained whenever both the above norm and the norm of the error in the first-order optimality conditions are less than their predetermined tolerances, > respectively. To facilitate numerical integrations, the interval of integration is normalized to unit length. Variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time T at which the terminal boundary is reached then becomes a parameter to be optimized. Two numerical examples substantiating the theory are presented.