Laplace II: Steps, Impulses & Convolution · Superchallenge problem

A damped oscillator starts at rest and is driven by a force of magnitude at most $1$: $y''+2y'+2y=f(t)$, $y(0)=y'(0)=0$, $|f(t)|\le1$. Over all such forces and all times $t\ge0$, find the least upper bound of $y(t)$.