Question $1$:

Consider the following problem

minJ$={\displaystyle \underset{0}{\overset{T}{}}}\frac{1}{2}u(t{)}^{2}dt$

subject to

$x^{}(t)=u(t)-t$

where $0tT$

a$)$ Augment the system dynamics by adding time as a new state variable, $t^{}=1$ to obtain a

time$-$invariant system. Present the state variables, new system dynamics, and boundary

conditions for the augmented system.

b$)$ Find the Hamiltonian, and write the costate equations, state equations, and optimal control

policy using the Pontryagin's Minimum Principle.

c$)$ Calculate the terminal time $T,$ and optimal control input by solving the equations presented

in part b$).$