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Given a piecewise smooth, bounded, integrable, etc., causal function $h(t)$, such that $h(t)=0$ if $t<0$, my question is which bounded, causal, piecewise smooth, etc., function $c(t)$ will maximize the magnitude $|R|$ of the convolution integral $R(t|c)=\int_{0}^{t}h(x)c(t-x)\,dx$ for all $t>0$ assuming that $|c(t)| \le 1$.

In other words, given the impulse response $h(t)$ of a causal "well-behaved" linear system I am looking for that bounded input signal $c(t)$ that gives the largest possible (in magnitude) output the system response can have.

If $h$ has finite support $h(t)=0$ for $t<0$ and $t>T$ then I think the answer is $\tilde {c} (t) =\text{sgn} (h(T-t))$ but a general case would still be interesting.

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