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The costate equation is related to the state equation used in optimal control.[1][2] It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector of first order differential equations

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

Interpretation

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The costate variables 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.[3][4]

Solution

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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.[5]

See also

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References

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  1. ^ Kamien, Morton I.; Schwartz, Nancy L. (1991). Dynamic Optimization (Second ed.). London: North-Holland. pp. 126–27. ISBN 0-444-01609-0.
  2. ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 263. ISBN 9780471181170.
  3. ^ Takayama, Akira (1985). Mathematical Economics. Cambridge University Press. p. 621. ISBN 9780521314985.
  4. ^ 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.
  5. ^ Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.

📚 Artikel Terkait di Wikipedia

Hamiltonian (control theory)

equations for the state variables), and the terminal time (the n {\displaystyle n} differential equations for the costate variables; unless a final function

Transversality condition

condition is a boundary condition for the terminal values of the costate variables. They are one of the necessary conditions for optimality infinite-horizon

Bond graph

use the concept of analogous power conjugate variables whose product is energy flow, or power; these variable pairs are called effort and flow and, for example

Value function

{\displaystyle \partial V(t,x)/\partial x=\lambda (t)} playing the role of the costate variables. Given this definition, we further have d λ ( t ) / d t = ∂ 2 V (

Ramsey–Cass–Koopmans model

\left[f(k)-(n+\delta )k-c\right]} where μ {\displaystyle \mu } is the costate variable usually economically interpreted as the shadow price. Because the terminal

Lagrange multiplier

optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of

Mechanism design

x} is a state variable and ∂ x / ∂ θ {\displaystyle \partial x/\partial \theta } the control. As usual in optimal control the costate evolution equation

Rabbit Semiconductor

costatements which will toggle our LEDs. costate { led1on(); waitfor(DelayMs(100)); led1off(); waitfor(DelayMs(50)); } costate { led2on(); waitfor(DelayMs(200));