References
While the indirect approaches to optimal control constitute the classical core of the optimal control theory, most treatments of the subject consider continuous-time systems. Our treatment was based on Chapter 2 in (Lewis, Vrabie, and Syrmo 2012), which is one of a few resources that discuss discrete-time optimal control too.
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Lewis, Frank L., Draguna Vrabie, and Vassilis L. Syrmo. 2012. Optimal Control. 3rd ed. John Wiley & Sons. https://lewisgroup.uta.edu/FL%20books/Lewis%20optimal%20control%203rd%20edition%202012.pdf.