Optimal control

Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized.^[1] It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure.^[2] Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy.^[3] A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory.^[4]^[5]

Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies.^[6] The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane.^[7] Optimal control can be seen as a control strategy in control theory.^[1]

^ ^a ^b Ross, Isaac (2015). A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. ISBN 978-0-9843571-0-9. OCLC 625106088.
^ Luenberger, David G. (1979). "Optimal Control". Introduction to Dynamic Systems. New York: John Wiley & Sons. pp. 393–435. ISBN 0-471-02594-1.
^ Kamien, Morton I. (2013). Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management. Dover Publications. ISBN 978-1-306-39299-0. OCLC 869522905.
^ Ross, I. M.; Proulx, R. J.; Karpenko, M. (6 May 2020). "An Optimal Control Theory for the Traveling Salesman Problem and Its Variants". arXiv:2005.03186 [math.OC].
^ Ross, Isaac M.; Karpenko, Mark; Proulx, Ronald J. (1 January 2016). "A Nonsmooth Calculus for Solving Some Graph-Theoretic Control Problems**This research was sponsored by the U.S. Navy". IFAC-PapersOnLine. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. 49 (18): 462–467. doi:10.1016/j.ifacol.2016.10.208. ISSN 2405-8963.
^ Sargent, R. W. H. (2000). "Optimal Control". Journal of Computational and Applied Mathematics. 124 (1–2): 361–371. Bibcode:2000JCoAM.124..361S. doi:10.1016/S0377-0427(00)00418-0.
^ Bryson, A. E. (1996). "Optimal Control—1950 to 1985". IEEE Control Systems Magazine. 16 (3): 26–33. doi:10.1109/37.506395.

[:0-1] Ross, Isaac (2015). A primer on Pontryagin's principle in optimal control. San Francisco: Collegiate Publishers. ISBN 978-0-9843571-0-9. OCLC 625106088.

[2] Luenberger, David G. (1979). "Optimal Control". Introduction to Dynamic Systems. New York: John Wiley & Sons. pp. 393–435. ISBN 0-471-02594-1.

[3] Kamien, Morton I. (2013). Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management. Dover Publications. ISBN 978-1-306-39299-0. OCLC 869522905.

[4] Ross, I. M.; Proulx, R. J.; Karpenko, M. (6 May 2020). "An Optimal Control Theory for the Traveling Salesman Problem and Its Variants". arXiv:2005.03186 [math.OC].

[5] Ross, Isaac M.; Karpenko, Mark; Proulx, Ronald J. (1 January 2016). "A Nonsmooth Calculus for Solving Some Graph-Theoretic Control Problems**This research was sponsored by the U.S. Navy". IFAC-PapersOnLine. 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016. 49 (18): 462–467. doi:10.1016/j.ifacol.2016.10.208. ISSN 2405-8963.

[6] Sargent, R. W. H. (2000). "Optimal Control". Journal of Computational and Applied Mathematics. 124 (1–2): 361–371. Bibcode:2000JCoAM.124..361S. doi:10.1016/S0377-0427(00)00418-0.

[7] Bryson, A. E. (1996). "Optimal Control—1950 to 1985". IEEE Control Systems Magazine. 16 (3): 26–33. doi:10.1109/37.506395.

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