Hamilton function (control theory)

The Hamilton function in the theory of optimal controls was developed by Lew Pontryagin as part of his maximum principle. It is similar to the Hamilton function of mechanics, but differs from it. Pontryagin showed that a necessary condition for solving an optimal control problem is that the chosen control must minimize the Hamilton function.

Notation and problem definition

A control should be chosen in such a way that the following objective functions are minimized ${\displaystyle u(t)}$

${\displaystyle J(u)=\Psi (x(T))+\int _{0}^{T}L(x,u,t)\mathrm {d} t}$

where describes the state of the system, which is according to the differential equations ${\displaystyle x(t)}$

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

developed, and the controller must meet the following restrictions ${\displaystyle u(t)}$

${\displaystyle a\leq u(t)\leq b\quad t\in [0,T].}$

Furthermore, there is any function of the target state according to time as well as the Lagrangian function , which describes the dynamics of the system under consideration. ${\displaystyle \Psi (x(T))}$ ${\displaystyle x(T)}$ ${\displaystyle T}$ ${\displaystyle L(x,u,t)}$

Definition of the Hamilton function

${\displaystyle H(x,\lambda ,u,t)=\lambda ^{T}(t)f(x,u,t)+L(x,u,t)\,}$

where are the Lagrange multipliers whose components describe the adjoint states. ${\displaystyle \lambda (t)}$

literature

• Velimir Jurdjevic: Geometric Control Theory (=  Cambridge Studies in Advanced Mathematics . Volume 52 ). Cambridge University Press, Cambridge u. a. 2008, ISBN 978-0-521-05824-7 .