Editing Costate equation

{{Short description|Optimal control equation}}
The '''costate equation''' is related to the state equation used in [[optimal control]].<ref>{{cite book |first1=Morton I. |last1=Kamien |author-link1=Morton Kamien |first2=Nancy L. |last2=Schwartz |title=Dynamic Optimization |location=London |publisher=North-Holland |edition=Second |year=1991 |isbn=0-444-01609-0 |pages=126–27 |url=https://books.google.com/books?id=0IoGUn8wjDQC&pg=PA126 }}</ref><ref>{{cite book |first=David G. |last=Luenberger |author-link=David Luenberger |title=Optimization by Vector Space Methods |location=New York |publisher=John Wiley & Sons |year=1969 |isbn= 9780471181170|page=263 |url=https://books.google.com/books?id=lZU0CAH4RccC&pg=PA263 }}</ref> It is also referred to as '''auxiliary''', '''adjoint''', '''influence''', or '''multiplier equation'''. It is stated as a [[vector (geometry)|vector]] of first order [[differential equation]]s
:<math>
\dot{\lambda}^{\mathsf{T}}(t)=-\frac{\partial H}{\partial x}
</math>
where the right-hand side is the vector of [[partial derivative]]s of the negative of the [[Hamiltonian (control theory)|Hamiltonian]] with respect to the state variables.

== Interpretation ==
The costate variables <math>\lambda(t)</math> can be interpreted as [[Lagrange multipliers]] associated with the state equations.  The state equations represent constraints of the minimization problem, and the costate variables represent the [[marginal cost]] of violating those constraints; in economic terms the costate variables are the [[shadow price]]s.<ref>{{cite book |first=Akira |last=Takayama |title=Mathematical Economics |publisher=Cambridge University Press |year=1985 |page=621 |url=https://books.google.com/books?id=j6PLOBFotPQC&pg=PA621 |isbn=9780521314985 }}</ref><ref>{{cite journal |first=Daniel |last=Léonard |title=Co-state Variables Correctly Value Stocks at Each Instant : A Proof |journal=Journal of Economic Dynamics and Control |volume=11 |issue=1 |year=1987 |pages=117–122 |doi=10.1016/0165-1889(87)90027-3 }}</ref>

== Solution ==
The state equation is subject to an initial condition and is solved forwards in time.  The costate equation must satisfy a [[transversality condition]] and is solved backwards in time, from the final time towards the beginning.  For more details see [[Pontryagin's maximum principle]].<ref>[[I. Michael Ross|Ross, I. M.]] ''A Primer on Pontryagin's Principle in Optimal Control'', Collegiate Publishers, 2009. {{ISBN|978-0-9843571-0-9}}.</ref>

== See also ==
* [[Adjoint equation]]
* [[Covector mapping principle]]
* [[Lagrange multiplier]]

== References ==
{{reflist}}

{{DEFAULTSORT:Costate Equation}}
[[Category:Optimal control]]
[[Category:Calculus of variations]]