Editing Optimal control (section)

==Discrete-time optimal control==
The examples thus far have shown [[continuous time]] systems and control solutions. In fact, as optimal control solutions are now often implemented [[Digital data|digital]]ly, contemporary control theory is now primarily concerned with [[discrete time]] systems and solutions.  The Theory of [[Consistent Approximations]]<ref>E. Polak, ''On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems'' Math. Prog. 62 pp. 385–415 (1993).</ref><ref>{{Cite journal|last=Ross|first=I M. |date=2005-12-01|title=A Roadmap for Optimal Control: The Right Way to Commute |url=http://dx.doi.org/10.1196/annals.1370.015 |journal=Annals of the New York Academy of Sciences|volume=1065 |issue=1 |pages=210–231 |doi=10.1196/annals.1370.015 |pmid=16510411 |bibcode=2005NYASA1065..210R |s2cid=7625851 |issn=0077-8923|url-access=subscription }}</ref> provides conditions under which solutions to a series of increasingly accurate discretized optimal control problem converge to the solution of the original, continuous-time problem.  Not all discretization methods have this property, even seemingly obvious ones.<ref>{{Cite journal|last1=Fahroo|first1=Fariba |last2=Ross|first2=I. Michael |date=September 2008|title=Convergence of the Costates Does Not Imply Convergence of the Control|url=http://dx.doi.org/10.2514/1.37331|journal=Journal of Guidance, Control, and Dynamics|volume=31|issue=5 |pages=1492–1497|doi=10.2514/1.37331 |bibcode=2008JGCD...31.1492F |s2cid=756939 |issn=0731-5090|hdl=10945/57005|hdl-access=free}}</ref>  For instance, using a variable step-size routine to integrate the problem's dynamic equations may generate a gradient which does not converge to zero (or point in the right direction) as the solution is approached.   The direct method ''[http://www.schwartz-home.com/RIOTS RIOTS]'' is based on the Theory of Consistent Approximation.