I have the following convolution equation:
$$ z(t) = u(t)*[0.5t + (t-1)H(t-1)], \ t \in [0\ 5], $$ where both functions $z(t)$ and $u(t)$ are unknown, $H(.)$ is the unit Heaviside function, and $*$ is the convolution operator. Moreover, the function $z(t)$ must satisfy the boundary conditions:
$$ z(5) = 1, \ \dot{z}(5) = 0. $$
My goal is: I would like to design the function $u(t)$ that satisfies the defined boundary values of $z(t)$.
It seems not difficult if we assume a special form of $u(t)$ (e.g. $a\sin(\omega t)$) and compute the parameters $a$ and $\omega$ by using the given boundary conditions of $z(t)$.
However, I would like to generalize the class of functions that $u(t)$ should belong to rather than a specific case.
So my question is: What are the special properties of $u(t)$ could we exploit from the above conditions? Or what tools we can use here?
I very much appreciate any suggestion!
Hint.
Using the Laplace transform we have
$$ Z(s) = U(s)\frac{2 e^{-s}+1}{2 s^2} $$
now making
$$ U(s) = \frac as+\frac{b}{s^2}+\frac{c}{s^3}\Rightarrow Z(s) = \left( \frac as+\frac{b}{s^2}+\frac{c}{s^3}\right)\frac{2 e^{-s}+1}{2 s^2} $$
and after anti transforming we obtain
$$ z(t) = \frac{1}{48} \left(2 (t-1)^2 H(t-1) (12 a+(t-1) (4 b+c (t-1)))+t^2 (12 a+t (4 b+c t))\right) $$
The determination of $a, b, c$ follows with the initial conditions:
$$ z(5) = 1\ \ \ z'(5) =0 $$
giving
$$ a = \frac{61609 c}{38028}\\ b = -\frac{14061 c}{6338} $$
This procedure can be applied for any set of $u(t)$ Laplace transformable functions.