The costate equation is related to the state equation used in

optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...

. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

of first order differential equations :

\dot^(t)=-\frac

where the right-hand side is the vector of

partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

s of the negative of the

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

with respect to the state variables.

Interpretation

The costate variables

\lambda(t)

can be interpreted as

Lagrange multipliers In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfie ...

associated with the state equations. The state equations represent constraints of the minimization problem, and the costate variables represent the

marginal cost In economics, the marginal cost is the change in the total cost that arises when the quantity produced is increased, i.e. the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it ...

of violating those constraints; in economic terms the costate variables are the

shadow price A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in ...

Solution

The state equation is subject to an initial condition and is solved forwards in time. The costate equation must satisfy a

transversality condition In optimal control theory, a transversality condition is a boundary condition for the terminal values of the costate variables. They are one of the necessary conditions for optimality infinite-horizon optimal control problems without an endpoint c ...

and is solved backwards in time, from the final time towards the beginning. For more details see

Pontryagin's maximum principle Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It states that it ...

. Ross, I. M. ''A Primer on Pontryagin's Principle in Optimal Control'', Collegiate Publishers, 2009. .

References

{{DEFAULTSORT:Costate Equation Optimal control Calculus of variations

Interpretation

Solution

See also

References