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
where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables.
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.
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.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
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 where the right-hand side is the vector of partial derivatives of the negative of the Hamiltonian with respect to the state variables. The costate variables can be interpreted as Lagrange multipliers associated with the state equations.
Optimal control theory is a branch of mathematical optimization 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 research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.
The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian.
This doctoral course provides an introduction to optimal control covering fundamental theory, numerical implementation and problem formulation for applications.
Covers the fundamentals of optimal control theory, focusing on defining OCPs, existence of solutions, performance criteria, physical constraints, and the principle of optimality.