Costate equation

<h2 id="interpretation">Interpretation</h2>
<p>The costate variables 
  
    
      
        λ
        (
        t
        )
      
    
    {\displaystyle \lambda (t)}
  
 can be interpreted as <a href="/facts/Lagrange_multipliers/U5mSI5YP">Lagrange multipliers</a> associated with the state equations.  The state equations represent constraints of the minimization problem, and the costate variables represent the <a href="/facts/Marginal_cost/fcKjM5Mr">marginal cost</a> of violating those constraints; in economic terms the costate variables are the <a href="/facts/Shadow_price/gpHIueEC">shadow prices</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="solution">Solution</h2>
<p>The state equation is subject to an initial condition and is solved forwards in time.  The costate equation must satisfy a <a href="/facts/Transversality_condition/QDE96E1u">transversality condition</a> and is solved backwards in time, from the final time towards the beginning.  For more details see <a href="/facts/Pontryagin%2527s_maximum_principle/gEVJ1sjc">Pontryagin's maximum principle</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Adjoint_equation/e6yG2P1m">Adjoint equation</a></li>
<li><a href="/facts/Covector_mapping_principle/QpoRMXFz">Covector mapping principle</a></li>
<li><a href="/facts/Lagrange_multiplier/U5mSI5YP">Lagrange multiplier</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Kamien, Morton I.; Schwartz, Nancy L. (1991). Dynamic Optimization (Second ed.). London: North-Holland. pp. 126–27. ISBN 0-444-01609-0. <a href="0-444-01609-0" target="_blank">0-444-01609-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 263. ISBN 9780471181170. <a href="9780471181170" target="_blank">9780471181170</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Takayama, Akira (1985). Mathematical Economics. Cambridge University Press. p. 621. ISBN 9780521314985. <a href="9780521314985" target="_blank">9780521314985</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Léonard, Daniel (1987). "Co-state Variables Correctly Value Stocks at Each Instant : A Proof". Journal of Economic Dynamics and Control. 11 (1): 117–122. doi:10.1016/0165-1889(87)90027-3. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9. <a href="/wiki/I._Michael_Ross" target="_blank">/wiki/I._Michael_Ross</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Costate equation open-in-new

Costate equation