Costate equations

Costate equations are related to the state equations used in optimal control. (Sometimes they are also referred to as the adjoint equations.) They are stated as a vector of first order differential equations with the right-hand side being the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.^[1]

$\dot{\lambda}(t)=-\frac{\partial H}{\partial x}$

Interpretation

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.

Solution

The state equations are subject to an initial condition and are solved forwards in time. The costate equations must satisfy a terminal condition and are solved backwards in time, from the final time towards the beginning. For more details see Pontryagin's minimum principle.^[2]

See Also

Covector mapping principle

References

1. Moritz Diehl: Pontryagin's Maximum Principle
2. Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.