I'm trying to follow a demontration written in an optimal control paper. In one of the steps, it states
What I'm having troubles with is the last step, it states that because of the convexity of the function (17) it is "clear" that the optimal control is that in the equation (20). However, it is not that clear for me :-(. How are the authors deriving the result in equation (20) from the previous steps (i.e. the convexity of the function in 17)? A link to the complete paper is:
Thanks in advance.
$u(N-1)$ is a single real variable, hence we can consider the problem to be of the form $\min_{|\nu| \le B } f(\nu)$, where $f$ is a convex function of a single variable.
Suppose $\hat{\nu}$ is an unconstrained minimum of $f$, then $f$ is convex, we see that it is non decreasing on $[\hat{\nu}, \infty)$ and non increasing on $(-\infty, \hat{\nu}]$.
Consequently, if $\hat{\nu} \in [-B,B]$, then $\hat{\nu}$ solves the problem, if $\hat{\nu} \in (B,\infty)$, then $B$ solves the problem and if $\hat{\nu} \in (-\infty,-B)$, then $-B$ solves the problem.
Combining these gives the solution as $\operatorname{sat}_B \hat{\nu}$.
In the above, this gives $u^0(N-1) = \operatorname{sat}_\bar{U} (-K(N-1)x(N-1) ) = - \operatorname{sat}_\bar{U} (K(N-1)x(N-1) )$ (since $\operatorname{sat}$ is odd).
Note: The above really depends on unimodality more than convexity.