Question $2$:

Consider the following problem

minJ$={\displaystyle \underset{0}{\overset{T}{}}}((x(t)-c_1{)}^2+(y(t)-c_2{)}^2)dt$

subject $to$

$x^{}(t)=x(t)(-y(t)-u_1(t))$

$y^{}(t)=-y(t)(-y(t)-u_2(t))$

where the model parameters $,,,$ are positive constants. $x_0,y_0>0$ are the initial conditions. $u_1(t),u_2(t)in[0,1]$ are control inputs. And $T$ is a given terminal time.

a$)$ Find the Hamiltonian, and write the costate equations, state equations, and optimal control policy using the Pontryagin's Minimum Principle.

b$)$ Assign some numerical values for $x_0,y_0,c_1,c_2,$ and solve the two$-$point boundary value problem obtained in part a$)$ via MATLAB. Visualize the optimal trajectories $x^{*}(t),y^{*}(t)$ and optimal control inputs $u_1^{*}(t),u_2^{*}(t)$