Given a single mass control system as in the picture shown with provided variables, please derive the following: $1$a$.$ the dynamic modeling for the system, $1$b$.$ please derive the discrete mode for the system, $1$c$.$ Through dynamic propagation, use the discrete model to derive the linear algebra equation, $1$d$.$ find the least square solution and derive U$0*,$ U$1*,$ U$2*,$ U$3*$ that minimizes $1/2($U$0^2+$U$1^2+$U$2^2+$U$3^3).$