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Costate equation

Optimal control equation

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated February 7, 2025

The costate equation is related to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations

{\dot {\lambda }}^{\mathsf {T}}(t)=-{\frac {\partial H}{\partial x}}

where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.

01Interpretation

The costate variables \lambda (t) 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 prices.

02Solution

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.

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Sources and credits

This article is adapted from the Wikipedia article “Costate equation”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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