Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is \[\xrightarrow{{U\left( s \right)}}\boxed{\frac{1}{s}}\xrightarrow{{Y\left( s \right)}}\]

The correct answer is $\boxed{\frac{{{t^2}}}{2}u\left( t \right)}$.

The transfer function of a system is the ratio of its output to its input, expressed as a function of the Laplace transform variable $s$. The Laplace transform of a unit step input is $U(s) = \frac{1}{s}$. The Laplace transform of the output of a system is the product of the transfer function and the Laplace transform of the input.

In this case, the transfer function is $\frac{1}{s}$, so the Laplace transform of the output is $\frac{1}{s} \cdot \frac{1}{s} = \frac{1}{s^2}$. The inverse Laplace transform of $\frac{1}{s^2}$ is $\frac{t^2}{2}$, so the output of the system is $\frac{t^2}{2}u(t)$.

Option A, $u(t)$, is the output of a system with a transfer function of $1$. Option B, $tu(t)$, is the output of a system with a transfer function of $s$. Option C, $\frac{{{t^2}}}{2}u\left( t \right)$, is the output of a system with a transfer function of $\frac{1}{s^2}$. Option D, $e^{-tu(t)}$, is the output of a system with a transfer function of $e^{-t}$.