Problem $1$: Conversion of Hard Constraints to Soft Constraints $-$ Analytic Solution $(60$ points$)$ In the last homework set, you were given the performance index J$=\_0^11/2$ u$^2$ d t and the state constraints $\_1=$x$\_2$ and $\_2=3$ u$+1,$ with x$\_1(0)=0$ and x$\_2(0)=0.$ Some students ask about the difference between having soft constraints versus hard constraints. So$,$ let's modify the problem where we change the hard constraint requirements x$\_1(1)=2$ and x$\_2(1)=1$ into soft constraints, resulting in a modified performance index J$=1/2$W$\_1($x$\_1$ f$-2)^2+1/2$W$\_2($x$\_2$ f$-1)^2+\_0^11/2$ u$^2$ d t$.$ The scale factors W$\_$l and W$\_2$ give us the ability to weight each constraint independently and influence the amount of control needed to accomplish the objective to a point where the final states might be 'good enough.$\text{'}$ Note that there are no longer any hard constraints, just soft ones. In order to see the change in the solution, solve this problem analytically by forming the Hamiltonian and implementing the necessary conditions for optimality. Leave the scale factors $($W$\_1.$ and $.$W$\_2)$ as variables until you come up with a system of two equations that relate the two final states and the scale factors. Then, set W$\_$l$=$W$\_2=1$ and solve for x$\_$l f and x$\_2$ f$.$ With those values, write the final equations for x$\_1($t$),$ x$\_2($t$),$ and u$($t$).$