Calculus of variation says that the minimization of J with constraint is equivalent to the minimization of J_a without any constraint.

Let $\boldsymbol{x}^*(k),\; \boldsymbol{x}^*(k+1), \; \boldsymbol{u}^*(k))$ and $\boldsymbol{\lambda}^*(k+1)$ represent the vectors corresponding to optimal trajectories. Thus one can write

where $\boldsymbol{\eta}(k), \; \boldsymbol{\mu}(k), \; \boldsymbol{\nu}(k)$ are arbitrary vectors and $\epsilon, \; \delta, \;\gamma$ are small constants.

Substituting the above four equations in the expression of J_a,

To simplify the notation, let us denote J_a as

$\displaystyle {J}_a = \sum\limits_{k = {0}}^{{N-1}}F_a(k, \boldsymbol{x}(k), \boldsymbol{x}(k+1), \boldsymbol{u}(k)), \boldsymbol{\lambda}(k+1))$

Expanding F_a using Taylor series around $\boldsymbol{x}^*(k),\; \boldsymbol{x}^*(k+1), \; \boldsymbol{u}^*(k))$ and $\boldsymbol{\lambda}^*(k+1)$ , we get

where

$\displaystyle F_a^*(k)=F_a(k, \boldsymbol{x}^*(k), \boldsymbol{x}^*(k+1), \boldsymbol{u}^*(k), \boldsymbol{\lambda}^*(k+1))$

The necessary condition for J_a to be minimum is