The optimal solution can be calculated with high-precision (see TP3_ClosedFormEquations). The control-only, inner-loop solution (i.e., the optimal control for any given k) can be solved using a linear program created with a direct transcription method (see TP3_Inner). In addition, the code is available to replicate the results from the papers below.
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Section 5.3 of DR Herber, JT Allison. Nested and Simultaneous Solution Strategies for General Combined Plant and Control Design Problems. ASME Journal of Mechanical Design (to appear), MD-18-1066, 2018. [bibtex]
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Section 3.1 of DR Herber, JT Allison. Unified Scaling of Dynamic Optimization Design Formulations. In ASME 2017 International Design Engineering Technical Conferences, DETC2017-67676, Cleveland, OH, USA, Aug 2017. [bibtex] [pdf]