Optimal control of nonlinear systems with unspecified final times

The use of time stages of varying lengths in iterative dynamic programming (IDP) allows the establishment of the optimal control of engineering systems where the final time is not specified, even when the system equations are nonlinear and constrained. The viability of this method is illustrated by studying the convergence characteristics of three typical nonlinear chemical engineering systems. By incorporating a penalty factor for the final time into the objective function, it is shown that the operating times of such problems can be reduced without significantly affecting the value of the performance index. The procedure is straightforward and the computations are readily carried out on a personal computer.