Hamiltonian (control theory)
The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. It was inspired by, but is distinct from, the Hamiltonian of classical mechanics. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to minimize the Hamiltonian. For details see Pontryagin's maximum principle.
Notation and Problem statementEdit
A control is to be chosen so as to minimize the objective function
where is the system state, which evolves according to the state equations
and the control must satisfy the constraints
Definition of the HamiltonianEdit
where is a vector of costate variables of the same dimension as the state variables .
For information on the properties of the Hamiltonian, see Pontryagin's maximum principle.
The Hamiltonian in discrete timeEdit
When the problem is formulated in discrete time, the Hamiltonian is defined as:
and the costate equations are
(Note that the discrete time Hamiltonian at time involves the costate variable at time  This small detail is essential so that when we differentiate with respect to we get a term involving on the right hand side of the costate equations. Using a wrong convention here can lead to incorrect results, i.e. a costate equation which is not a backwards difference equation).
The Hamiltonian of control compared to the Hamiltonian of mechanicsEdit
where the Lagrangian the extremizing of which determines the dynamics (not the Lagrangian defined above), is the state variable and is its time derivative.
is the so-called "conjugate momentum", defined by
Hamilton then formulated his equations to describe the dynamics of the system as
The Hamiltonian of control theory describes not the dynamics of a system but conditions for extremizing some scalar function thereof (the Lagrangian) with respect to a control variable . As normally defined, it is a function of 4 variables
where is the state variable and is the control variable with respect to which we are extremizing.
The associated conditions for a maximum are
This definition agrees with that given by the article by Sussmann and Willems. (see p. 39, equation 14). Sussmann-Willems show how the control Hamiltonian can be used in dynamics e.g. for the brachystochrone problem, but do not mention the prior work of Carathéodory on this approach.
Example: Ramsey ModelEdit
Take a simplified version of the Ramsey–Cass–Koopmans model. We wish to maximize an agent's discounted lifetime utility achieved through consumption
subject to the time evolution of capital per effective worker
where is period t consumption, is period t capital per worker, is period t production, is the population growth rate, is the capital depreciation rate, the agent discounts future utility at rate , with and .
Here, is the state variable which evolves according to the above equation, and is the control variable. The Hamiltonian becomes
The optimality conditions are
If we let , then log-differentiating the first optimality condition with respect to yields
Inserting this equation into the second optimality condition yields
- Dixit, Avinash K. (1990). Optimization in Economic Theory. New York: Oxford University Press. pp. 145–161. ISBN 0-19-877210-6.
- Varaiya, Chapter 6
- Sussmann; Willems (June 1997). "300 Years of Optimal Control" (PDF). IEEE Control Systems.
- See Pesch, H. J.; Bulirsch, R. (1994). "The maximum principle, Bellman's equation, and Carathéodory's work". Journal of Optimization Theory and Applications. 80 (2): 199–225. doi:10.1007/BF02192933.