A computational technique for optimal control problems with state variable constraint

Abstract

where F, f, and g are piecewise C z in the (x, u) space. Physically, such problems arise from attempting to control a dynamic system from an initial state c to some terminal state x(T) so that some performance criterion F is minimized and the motion of the dynamic system lies within some clased surface g in the state space. Such problems have received considerable attention recently. Dreyfus and Gamkrelidze studied the necessary conditions for optimality for problem (P-l) [1, 2]. Berkovitz, Desoer, and Pontriagin et al. treated similar problems under the now well known heading of the Maximum Principle [3-5]. However, effective computational techniques for solving such problems were not developed until recently. Bryson and Kelley first proposed a successive approximation procedure for solving such problems without any inequality constraint [6, 7]. Dreyfus and Bryson then extended the technique to include (P-l) [8, 9]. Ho and Neustadt also developed computational techniques for the case of linear dynamic system with inequality constraint on the decision variable u [10, 11]. The present paper demonstrates another computational technique for (P-I) which is an extension of the method presented in ref. 10.