When the objective is to control the system in such a way that the control input is not too large, the corresponding performance index is

$\displaystyle {J_3} = \sum\limits_{k = {N_0}}^{{N_f}} {{\boldsymbol{u}^T}(k)\boldsymbol{u}(k)}$

Or,

$\displaystyle {J_4} = \sum\limits_{k = {N_0}}^{{N_f}} {{\boldsymbol{u}^T}(k)R\boldsymbol{u}(k)}$

where the weight matrix R is symmetric positive definite.

We cannot simultaneously minimize the performance indices J₁ and J₃ because minimization of J₁ requires large control input whereas minimization of J₃ demands a small control. A compromise between the two conflicting objects is

$\displaystyle J_5 = \lambda J_1 + (1-\lambda) J_3$

$\displaystyle \;\;\;\; \;\; \;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;... ...k)\boldsymbol{x}(k) + (1-\lambda)\boldsymbol{u}^T(k)\boldsymbol{u}(k) \right ]$

A generalization of the above performance index is

$\displaystyle J_6 = \sum\limits_{k = {N_0}}^{{N_f}} {\left[ {{\boldsymbol{x}^T}(k)Q\boldsymbol{x}(k) + \boldsymbol{u}^T(k)R\boldsymbol{u}(k)} \right]}$

which is the most commonly used quadratic performance index.

In certain applications, we may wish the final state to be close to 0 . Then a suitable performance index is

$\displaystyle {J_7} = {\boldsymbol{x}^T}({N_f})F\boldsymbol{x}({N_f})$

When the control objective is to keep the state small, the control input not too large and the final state as close to 0 as possible, we can combine J₆ and J₇, to get the most general performance index

$\displaystyle J_8 = \frac{1}{2}{\boldsymbol{x}^T}({N_f})F\boldsymbol{x}({N_f}) ... ...l{x}^T}(k)Q\boldsymbol{x}(k) + \boldsymbol{u}^T(k)R\boldsymbol{u}(k)} \right]}$

1/2 is introduced to simplify the manipulation.

Sometimes we want the system state to track a desired trajectory throughout the interval $\left[ {{N_0},{N_f}} \right]$ . In that case the performance index J₈ can be modified as