Optimization of Dynamic Systems

Consider a mass$-$spring$-$damper system described by the state$-$space

$x_1^{}(t)=x_2(t)$

$x_2^{}(t)=-\frac{k}{m}{x}_1(t)-\frac{b}{m}{x}_2(t)+\frac{1}{m}u(t)$

where, $x_1(t)$ is the position of the mass, $x_{-}(2)(t)$ is its velocity, $u(t)$ is the control force, $k=1$ is

the spring constant, $b=1$ is the damping constant, and $m=1$ is the mass of the system.

The objective is to drive the mass from an initial position $x_1(0)=0$ to a final position $x_1(3)=1$

while minimizing the following cost functional

$J(x(t),u(t))=\frac{1}{2}{\displaystyle \underset{0}{\overset{T}{}}}x(t{)}^2+u(t{)}^{2}dt$

$($a$)$ Formulate the Hamiltonian!

$($b$)$ Formulate the first$-$ and second$-$order necessary conditions $($canonical differential

equation, coupling equation and Legendre condition$)!$

Now the mass should be moved to the final position $x(1)(T)=1$ in shortest possible time.

Except from adhering to the input constraints, the control effort should not be considered in

the optimization problem.

$($c$)$ Formulate the cost functional!

$($d$)$ Formulate the Hamiltonian!

Hint: The system is linear!

$($e$)$ How will the optimal input trajectoy u$*$ basically look like? Answer intuitively!

Which theoretical statements can be made about this solution?

$($f$)$ Sketch the solution of the time$-$optimal control problem in the x$1-$x$2-$plane!