Proof: Equation (1) can also be represented as

$\displaystyle \Delta V(\boldsymbol{x}(k))\vert _{\boldsymbol{u}=\boldsymbol{u}^... ...ymbol{x}^T(k)Q \boldsymbol{x}(k) + {\boldsymbol{u}^*}^T(k)R\boldsymbol{u}^*(k)$

Hence, we can write

$\displaystyle \Delta V(\boldsymbol{x}(k))\vert _{\boldsymbol{u}=\boldsymbol{u}^... ...mbol{x}^T(k)Q \boldsymbol{x}(k) - {\boldsymbol{u}^*}^T(k)R\boldsymbol{u}^*(k)$
We can sum both sides of the above equation from 0 to ∞ and get

$\displaystyle V(\boldsymbol{x}(\infty))-V(\boldsymbol{x}(0))= -\sum\limits_{k =... ...ymbol{x}^T(k)Q\boldsymbol{x}(k) + {\boldsymbol{u}^*}^T(k)R\boldsymbol{u}^*(k))$

Since the closed loop system is stable by assumption, $\boldsymbol{x}(\infty)=0$ and hence $V(\boldsymbol{x}(\infty))=0$ . Thus

$\displaystyle V(\boldsymbol{x}(0))= \sum\limits_{k = 0}^\infty (\boldsymbol{x}^T(k)Q\boldsymbol{x}(k) + {\boldsymbol{u}^*}^T(k)R\boldsymbol{u}^*(k))$

Now, $V(\boldsymbol{x}(0))=\boldsymbol{x}_o^TP\boldsymbol{x}_0$ .

Thus if a linear state feedback controller satisfies the hypothesis of the theorem the value of the resulting cost function is

$\displaystyle J(\boldsymbol{u}^*) = \boldsymbol{x}_o^TP\boldsymbol{x}_0$

To show that such a controller is indeed optimal, we will use a proof by contradiction.

Assume that the hypothesis of the theorem holds true but the controller is not optimal. Thus there exists a controller $\bar {\boldsymbol{u}}$ such that

$\displaystyle J(\bar {\boldsymbol{u}}) < J(\boldsymbol{u}^*)$

Using the theorem, we can write

$\displaystyle \Delta V(\boldsymbol{x}(k))\vert _{\boldsymbol{u}=\bar {\boldsymb... ...oldsymbol{x}(k) + {\bar {\boldsymbol{u}}}^T(k)R\bar {\boldsymbol{u}}(k) \ge 0$

The above can be rewritten as

$\displaystyle \Delta V(\boldsymbol{x}(k))\vert _{\boldsymbol{u}=\bar{ \boldsymb... ...(k)Q \boldsymbol{x}(k) - {\bar {\boldsymbol{u}}}^T(k)R\bar {\boldsymbol{u}}(k)$

Summing the above from 0 to ∞,

$\displaystyle V(\boldsymbol{x}(0)) \le \sum\limits_{k = 0}^\infty (\boldsymbol{... ...(k)Q\boldsymbol{x}(k) + {\bar{\boldsymbol{u}}}^T(k)R \bar{ \boldsymbol{u}}(k))$

The above inequality implies that $\displaystyle J(\boldsymbol{u}^*) \le J(\bar{\boldsymbol{ u}})$
which is a contradiction of our earlier assumption. Thus u* is optimal.

For more details one may consult Systems and Control by Stanislaw H. Zak