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<h2 id="further-reading">Further reading</h2> <ul><li>Caputo, Michael R. (2005). <a href="https://books.google.com/books?id=XZ2yYSVKWJkC&pg=PA174">"Necessary and Sufficient Conditions for Isoperimetric Problems"</a>. <i>Foundations of Dynamic Economic Analysis : Optimal Control Theory and Applications</i>. New York: Cambridge University Press. pp. 174–210. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-60368-4.</li> <li>Clarke, Frank H.; Loewen, Philip D. (1986). "The Value Function in Optimal Control: Sensitivity, Controllability, and Time-Optimality". <i>SIAM Journal on Control and Optimization</i>. 24 (2): 243–263. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F0324014">10.1137/0324014</a>.</li> <li>LaFrance, Jeffrey T.; Barney, L. Dwayne (1991). <a href="http://ageconsearch.umn.edu/record/259398/files/agecon-montanastate-003.pdf">"The Envelope Theorem in Dynamic Optimization"</a> (PDF). <i>Journal of Economic Dynamics and Control</i>. 15 (2): 355–385. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0165-1889%2891%2990018-V">10.1016/0165-1889(91)90018-V</a>.</li> <li>Stengel, Robert F. (1994). <a href="https://books.google.com/books?id=jDjPxqm7Lw0C&pg=PA201">"Conditions for Optimality"</a>. <i>Optimal Control and Estimation</i>. New York: Dover. pp. 201–222. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-68200-5.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Fleming, Wendell H.; Rishel, Raymond W. (1975). Deterministic and Stochastic Optimal Control. New York: Springer. pp. 81–83. ISBN 0-387-90155-8. <a href="0-387-90155-8" target="_blank">0-387-90155-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Caputo, Michael R. (2005). Foundations of Dynamic Economic Analysis : Optimal Control Theory and Applications. New York: Cambridge University Press. p. 185. ISBN 0-521-60368-4. <a href="0-521-60368-4" target="_blank">0-521-60368-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weber, Thomas A. (2011). Optimal Control Theory : with Applications in Economics. Cambridge: The MIT Press. p. 82. ISBN 978-0-262-01573-8. <a href="978-0-262-01573-8" target="_blank">978-0-262-01573-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bertsekas, Dimitri P.; Tsitsiklis, John N. (1996). Neuro-Dynamic Programming. Belmont: Athena Scientific. p. 2. ISBN 1-886529-10-8. <a href="1-886529-10-8" target="_blank">1-886529-10-8</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"EE365: Dynamic Programming" (PDF). <a href="https://stanford.edu/class/ee365/lectures/dp.pdf#page=3" target="_blank">https://stanford.edu/class/ee365/lectures/dp.pdf#page=3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Theory. New York: Oxford University Press. p. 964. ISBN 0-19-507340-1. <a href="0-19-507340-1" target="_blank">0-19-507340-1</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Corbae, Dean; Stinchcombe, Maxwell B.; Zeman, Juraj (2009). An Introduction to Mathematical Analysis for Economic Theory and Econometrics. Princeton University Press. p. 145. ISBN 978-0-691-11867-3. <a href="978-0-691-11867-3" target="_blank">978-0-691-11867-3</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Kamien, Morton I.; Schwartz, Nancy L. (1991). Dynamic Optimization : The Calculus of Variations and Optimal Control in Economics and Management (2nd ed.). Amsterdam: North-Holland. p. 259. ISBN 0-444-01609-0. <a href="0-444-01609-0" target="_blank">0-444-01609-0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Bertsekas, Dimitri P.; Tsitsiklis, John N. (1996). Neuro-Dynamic Programming. Belmont: Athena Scientific. p. 2. ISBN 1-886529-10-8. <a href="1-886529-10-8" target="_blank">1-886529-10-8</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Ljungqvist, Lars; Sargent, Thomas J. (2018). Recursive Macroeconomic Theory (Fourth ed.). Cambridge: MIT Press. p. 106. ISBN 978-0-262-03866-9. <a href="978-0-262-03866-9" target="_blank">978-0-262-03866-9</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Benveniste and Scheinkman established sufficient conditions for the differentiability of the value function, which in turn allows an application of the envelope theorem, see Benveniste, L. M.; Scheinkman, J. A. (1979). "On the Differentiability of the Value Function in Dynamic Models of Economics". Econometrica. 47 (3): 727–732. doi:10.2307/1910417. JSTOR 1910417. Also see Seierstad, Atle (1982). "Differentiability Properties of the Optimal Value Function in Control Theory". Journal of Economic Dynamics and Control. 4: 303–310. doi:10.1016/0165-1889(82)90019-7. <a href="/wiki/Jos%C3%A9_Scheinkman" target="_blank">/wiki/Jos%C3%A9_Scheinkman</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Kirk, Donald E. (1970). Optimal Control Theory. Englewood Cliffs, NJ: Prentice-Hall. p. 88. ISBN 0-13-638098-0. <a href="0-13-638098-0" target="_blank">0-13-638098-0</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Zhou, X. Y. (1990). "Maximum Principle, Dynamic Programming, and their Connection in Deterministic Control". Journal of Optimization Theory and Applications. 65 (2): 363–373. doi:10.1007/BF01102352. S2CID 122333807. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Theorem 10.1 in Bressan, Alberto (2019). "Viscosity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems" (PDF). Lecture Notes. <a href="http://personal.psu.edu/axb62/PSPDF/HJlnotes19.pdf#page=54" target="_blank">http://personal.psu.edu/axb62/PSPDF/HJlnotes19.pdf#page=54</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Kamalapurkar, Rushikesh; Walters, Patrick; Rosenfeld, Joel; Dixon, Warren (2018). "Optimal Control and Lyapunov Stability". Reinforcement Learning for Optimal Feedback Control: A Lyapunov-Based Approach. Berlin: Springer. pp. 26–27. ISBN 978-3-319-78383-3. <a href="978-3-319-78383-3" target="_blank">978-3-319-78383-3</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>