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   "source": [
    "# Problem 1. [50 points] Moon Landing"
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    "Consider the vertical (1D) descent of a spacecraft onto the lunar surface. The descent phase starts at $t=0$, and the touchdown happens at (free) final time $t=T$. Because of the lack of atomosphere, the spacecraft needs to use thrust to negate free fall due to gravity, and thereby ensure soft landing. \n",
    "\n",
    "<img src=\"MoonLanding.png\">\n",
    "\n",
    "Let $g$ be the constant (lunar) acceleration due to gravity during the descent. Suppose the mass of the fuel at time $t$ is $m(t)$, which changes over $0\\leq t \\leq T$ since the fuel is burnt to generate the thrust $\\tau = -k\\dot{m}$, where the velocity of exhaust gas w.r.t. the spacecraft is a constant $k>0$. Let $h(t)$ be the altitude of the spacecraft above the lunar surface at time $t$. From Newton's law (force balance), we have\n",
    "\n",
    "$$m\\ddot{h} = \\tau - mg = -k\\dot{m} - mg.$$\n",
    "\n",
    "The initial conditions are given: $h(0):=h_{0}>0$, $\\dot{h}(0)=v_{0}<0$, $m(0)=m_{0}>0$. The terminal constraints are: $h(T)=0$, $\\dot{h}(T)=0$; however $m(T)$ is free. Here, the control $u$ is the rate of fuel consumption, i.e., $\\dot{m} = u$. The engine imposes the constraint: $-\\alpha \\leq u \\leq 0$, for some known constant $\\alpha>0$.\n",
    "\n",
    "Since the spacecraft would like to return to Earth, the control objective is to maximize the fuel remaining $m(T)$ at touchdown, i.e., to minimize the fuel consumed: $m_{0} - m(T)$."
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   "source": [
    "## (a) [3 + 2 + 5 = 10 points] Optimal Control Problem"
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    "(a.1) Clearly define the state vector $x = (x_{1}, x_{2}, x_{3})^{\\top}\\in\\mathbb{R}^{3}$ and write the optimal control problem in standard **Mayer form**.\n",
    "\n",
    "(a.2) Rewrite the same optimal control problem in **Lagrange form**. \n",
    "\n",
    "(a.3) Prove that the above optimal control problem is equivalent to minimizing $T$, i.e., **the most fuel economic descent is also the fastest descent**. "
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   "source": [
    "## (b) [(1 + 1 + 2 + 1) + 15 + 2 + 8  = 30 points] Analytical Solution"
   ]
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    "(b.1) For the optimal control problem in part (a.3), write the **Hamiltonian** $H$, the **costate ODEs**, the **PMP**, and the **transversality condition**.\n",
    "\n",
    "(b.2) **Prove that** there does not exist any subinterval of $[0,T]$ where the optimal control is singular.\n",
    "\n",
    "[**Hint:** If possible, assume that there exist a finite subinterval $[t_{1},t_{2}]$ with $0< t_{1} < t_{2} < T$, where the optimal control is singular. Then prove by contradiction. Use the physical condition that $x_{3}(t)>0$ for all $t\\in[0,T]$, even though we did not explicitly enforce that as part of the necessary condition.]\n",
    "\n",
    "(b.3) Using (b.1)-(b.2), **argue that** the optimal control $u^{*}$ must take extreme values, i.e., $u^{*}(t)\\in\\{-\\alpha,0\\}$ for all $t\\in[0,T]$. However, it is not yet clear how many switchings are possible.\n",
    "\n",
    "(b.4) It is rather technical to prove that at most one switching is possible in this problem, i.e., $u^{*}$ is bang-bang. We will not prove this. Instead, use this information to **show that** the optimal control policy is to **either** use full thrust from $t=0$ to $t=T$, **or** to use a period of zero thrust (free fall) followed by full thrust until touchdown, depending on certain switching curve in the state space."
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   "source": [
    "## (c) [10 points] Numerical Solution"
   ]
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   "source": [
    "In this exercise, we will use the direct optimal control solver ICLOCS (http://www.ee.ic.ac.uk/ICLOCS/default.htm) within MATLAB environment to numerically solve the problem in (a.3). Follow download instructions here: http://www.ee.ic.ac.uk/ICLOCS/Downloads.html (requires IPOPT; we do not recommend fmincon). Check out the \"Example Problems\" tab and its dropdown list. Solve a listed simple example problem to reproduce the plots in the website before coding up our problem in (a.3). Plot the **optimal control and optimal state trajectories** for (a.3) via ICLOCS. **Does your numerical solution corroborate the analytical findings in part (b)?** "
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