Table 1 Minimization problem: inf _s W(s)

Scalar minimization problem

infs W(s)

Potential

W(s)=a s²+b s+c|s| (c>0, a>0)

Solution

$\underset{s}{inf}W\left(s\right)=-\frac{1}{4a}{(\left|b\right|-c)}_{+}^2$

Minimizer

$s=-\frac{1}{2a}{(\left|b\right|-c)}_{+}\text{sign}\left(b\right)$

Abbreviations

${(\left|b\right|-c)}_{+}=\left\{\begin{array}{ll}0& \text{for}\left|b\right|\le c\\ \left|b\right|-c& \text{for}\left|b\right|>c\end{array}\right.$

$\text{sign}\left(b\right)=\frac{\left|b\right|}{b}\text{for}b\ne 0$