Editing Optimal control (section)

{{Short description|Mathematical way of attaining a desired output from a dynamic system}}
[[File:Optimal Control Luus.png|thumb|Optimal control problem benchmark (Luus) with an integral objective, inequality, and differential constraint]]

'''Optimal control theory''' is a branch of [[control theory]] that deals with finding a [[Control (optimal control theory)|control]] for a [[dynamical system]] over a period of time such that an [[objective function]] is optimized.<ref name=":0">{{Cite book|last=Ross|first=Isaac |title=A primer on Pontryagin's principle in optimal control|publisher=Collegiate Publishers|year=2015|isbn=978-0-9843571-0-9|location=San Francisco|oclc=625106088}}</ref> 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.<ref>{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Introduction to Dynamic Systems |url=https://archive.org/details/introductiontody00luen_582 |url-access=limited |location=New York |publisher=John Wiley & Sons |year=1979 |isbn=0-471-02594-1 |chapter=Optimal Control |pages=[https://archive.org/details/introductiontody00luen_582/page/n406 393]–435 }}</ref> Or the dynamical system could be a nation's [[economy]], with the objective to minimize [[unemployment]]; the controls in this case could be [[Fiscal policy|fiscal]] and [[monetary policy]].<ref>{{Cite book|author=Kamien, Morton I.| url=http://worldcat.org/oclc/869522905 |title=Dynamic Optimization: the Calculus of Variations and Optimal Control in Economics and Management|date=2013 |publisher=Dover Publications |isbn=978-1-306-39299-0|oclc=869522905}}</ref> A dynamical system may also be introduced to embed [[Operations research|operations research problems]] within the framework of optimal control theory.<ref>{{cite arXiv |last1=Ross|first1=I. M. |last2=Proulx|first2=R. J. |last3=Karpenko|first3=M. |date=2020-05-06 | title=An Optimal Control Theory for the Traveling Salesman Problem and Its Variants |class=math.OC|eprint=2005.03186}}</ref><ref>{{Cite journal|last1=Ross|first1=Isaac M.|last2=Karpenko|first2=Mark|last3=Proulx|first3=Ronald J.|date=2016-01-01|title=A Nonsmooth Calculus for Solving Some Graph-Theoretic Control Problems**This research was sponsored by the U.S. Navy.| journal=IFAC-PapersOnLine |series=10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016|language=en|volume=49| issue=18|pages=462–467| doi=10.1016/j.ifacol.2016.10.208| issn=2405-8963| doi-access=free}}</ref>

Optimal control is an extension of the [[calculus of variations]], and is a mathematical optimization method for deriving [[control theory|control policies]].<ref>{{cite journal |first=R. W. H. |last=Sargent |author-link=Roger W. H. Sargent |title=Optimal Control |journal=Journal of Computational and Applied Mathematics |volume=124 |issue=1–2 |year=2000 |pages=361–371 |doi=10.1016/S0377-0427(00)00418-0 |bibcode=2000JCoAM.124..361S |doi-access=free }}</ref> 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]].<ref>{{cite journal |first=A. E. |last=Bryson |author-link=Arthur E. Bryson |year=1996 |title=Optimal Control—1950 to 1985 |journal= IEEE Control Systems Magazine |volume=16 |issue=3 |pages=26–33 |doi=10.1109/37.506395 }}</ref> Optimal control can be seen as a [[control strategy]] in [[control theory]].<ref name=":0" />