The control-only, inner-loop solution is available (i.e., the optimal control is known for any given k). The objective function (as a function of k) is known but many local minimums can be present (see references below). See TP2_ClosedFormEquations for the closed-form equations. In addition, the code is available to replicate the results from the papers below.
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Section 5.2 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 3.2 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]
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pp. 166-167 of AE Bryson Jr and Y-C Ho, Applied Optimal Control. Taylor & Francis, 1975, isbn: 9780891162285.