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Hamiltonian (control theory)

The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle.[1] 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  [2] 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

William Rowan Hamilton defined the Hamiltonian for describing the mechanics of a system. It is a function of three variables:


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.[3] (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.[4]

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


which is the Keynes–Ramsey rule or the Euler–Lagrange equation, which gives a condition for consumption in every period which, if followed, ensures maximum lifetime utility.

See alsoEdit


  1. ^ Dixit, Avinash K. (1990). Optimization in Economic Theory. New York: Oxford University Press. pp. 145–161. ISBN 0-19-877210-6.
  2. ^ Varaiya, Chapter 6
  3. ^ Sussmann; Willems (June 1997). "300 Years of Optimal Control" (PDF). IEEE Control Systems.
  4. ^ 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.

External linksEdit