Hamiltonian (control Theory) - The Hamiltonian in Discrete Time

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

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

(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).

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