# Hamiltonian (control theory)

The Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his minimum 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 minimum principle.

## Notation and Problem statement

A control $u(t)$ is to be chosen so as to minimize the objective function

$J(u)=\Psi(x(T))+\int^T_0 L(x,u,t) dt$

where $x(t)$ is the system state, which evolves according to the state equations

$\dot{x}=f(x,u,t) \qquad x(0)=x_0 \quad t \in [0,T]$

and the control must satisfy the constraints

$a \le u(t) \le b \quad t \in [0,T]$

## Definition of the Hamiltonian

$H(x,\lambda,u,t)=\lambda^T(t)f(x,u,t)-L(x,u,t) \,$

where $\lambda(t)$ is a vector of costate variables of the same dimension as the state variables $x(t)$.

For information on the properties of the Hamiltonian, see Pontryagin's maximum principle.

## The Hamiltonian in discrete time

When the problem is formulated in discrete time, the Hamiltonian is defined as:

$H(x,\lambda,u,t)=\lambda^T(t+1)f(x,u,t)-L(x,u,t) \,$

and the costate equations are

$\lambda(t+1)=-\frac{\partial H}{\partial x}dt + \lambda(t)$

(Note that the discrete time Hamiltonian at time $t$ involves the costate variable at time $t+1.$[2] This small detail is essential so that when we differentiate with respect to $x$ we get a term involving $\lambda(t+1)$ 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 mechanics

William Rowan Hamilton defined the Hamiltonian as a function of three variables:

$\mathcal{H} = \mathcal{H}(p,q,t) = \langle p,\dot{q} \rangle -L(q,\dot{q},t)$

where $\dot{q}$ is defined implicitly by

$p = \frac{\partial L}{\partial \dot{q}}$

Hamilton then formulated his equations as

$\frac{ d}{ dt}p(t) = -\frac{\partial}{\partial q}\mathcal{H}$
$\frac{ d}{ dt}q(t) =~~\frac{\partial}{\partial p}\mathcal{H}$

Similarly the Hamiltonian of control theory (as normally defined) is a function of 4 variables

$H(q,u,p,t)= \langle p,\dot{q} \rangle -L(q,u,t)$

and the associated conditions for a maximum are

$\frac{dp}{dt} = -\frac{\partial H}{\partial q}$
$\frac{dq}{dt} = ~~\frac{\partial H}{\partial p}$
$\frac{\partial H}{\partial u} = 0$

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]

## References

1. ^ I. M. Ross A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009.
2. ^ Varaiya, Chapter 6
3. ^ Sussmann; Willems (June 1997). "300 Years of Optimal Control" (PDF). IEEE Control Systems.
4. ^ See H. J. Pesch- R. Bulirsch: J.O.T.A. 80 1994 199-225