LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Chebyshev Polynomials

Required data
A univariate Polynomial whose SOSINEQ range specification is desired

The Problem
Let $$p_n (x)$$ be a univariate polynomial of degree $$n, with \gamma$$ being the coefficient of $$x^n$$.

$$ \max_\gamma \gamma $$, subject to: $$|p_n(x)|\leq 1, \forall x \in [-1,1]$$.

Formulation
The absolute value constraint can be easily rewritten using two inequalities, namely: $$ \begin{align} 1+p_n(x) \geq 0 \\ 1-p_n(x) \geq 0, \\ \forall x \in [-1,1] \end{align} $$.

Solution
The optimal solution is $$\gamma^* = 2 ^{n-1}$$, with $$p^*_n (x) = arccos(cos nx)$$ being the n-th Chebyshev polynomial of the first kind.

Example
Code example in SOStools probably worthwhile!

Reference
SOS tools manual: https://sums-of-squares.github.io/sos/

Related LMIs
SOS Basics