How to charge a capacitor using repeatable damped sinusoid current signals?
I have simulated it Matlab/Simulink but it yields a weird behaviour
when the input is a damped sinusoid \$\color{red}{\text{current}}\$ signal

You are using a \$\color{red}{\text{current}}\$ signal and that will inevitably keep putting charge into the storage capacitor. Just think about what a capacitor does when you apply a current source to it: -
$$I = C\dfrac{dv}{dt}\longrightarrow V = \dfrac{1}{C}\int i.dt$$
Voltage rises to infinity.