Nested Direct Transcription Optimization for Singular Optimal Control Problems

Singular optimal control problems arise frequently on a broad range of chemical engineering applications. Determination of the accurate dynamic structure of the optimal solution profile and the junctions between optimal nonsingular and singular arcs is essential for operating strategies, as well as equipment designs for many processes. In a previous study (Chen, Shao, and Biegler, AIChE J. 60(3), pp. 966-979, 2014) a nested optimization formulation that finds the optimal mesh distribution and determines exact control profiles for nonsingular optimal control problems without state constraints is developed. This study extends this approach to singular optimal control problems. To satisfy the necessary optimality conditions for singular optimal control problems with a well-defined solution strategy, the overall nonlinear programming formulation resulting from direct transcription is decomposed into inner and outer problems. The key feature of this algorithm is that it converges to a solution that satisfies the discretized Euler-Lagrange equations of the original singular optimal control problem; we prove this under suitable assumptions. This is obtained through the introduction of pseudomultipliers that reconstruct the necessary optimality conditions for singular optimal control in the outer problem. We demonstrate this approach on eight classical singular control problems with known solutions, as well as three larger singular control problems derived from chemical engineering applications. (C) 2016 American Institute of Chemical Engineers AIChE J, 62: 3611-3627, 2016