Question

# Describe some optimal control problems in your major field or in another area of interest to you. How difficult would it be to find the optimal controls?

Solution

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$\textbf{A moon lander}$ is an optimal control problem which asks us how to bring a spacecraft to the moon's surface, using as little fuel as possible?

We are going to use a following notation:

\begin{align*}&h(t) - \color{#4257b2}\text{height at time t}\\& v(t) = h'(t) -\color{#4257b2}\text{velocity }\\& m(t) - \color{#4257b2}\text{mass of spacecraft(changing as fuel is burned)}\\& \alpha(t) -\color{#4257b2}\text{ thrust at time t}\end{align*}

where $\alpha\in[0,1]$.

From Newton's law we get

\begin{align*}m \ddot{h}=-g m+\alpha\end{align*}

where the right hand side being the difference of the gravitational force and the thrust of the rocket.

This equation can be modeled as a system:

\begin{align*}\left\{\begin{aligned} \dot{v}(t) &=-g+\frac{\alpha(t)}{m(t)} \\ \dot{h}(t) &=v(t) \\ \dot{m}(t) &=-k \alpha(t) \end{aligned}\right.\end{align*}

These equations can be represented as

\begin{align*}\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t), \alpha(t))\end{align*}

where $\mathbf{x}(t)=(v(t), h(t), m(t))$.

Since we want to minimize the amount of fuel used up we have to maximize the amount remaining once we have landed, so we get the function

\begin{align*}P[\alpha(\cdot)]=m(y)\end{align*}

where $y$ is the value which represents the first time that $h(y)=v(y)=0$.

We have a problem here because the deadline is not predetermined, so we have the following conditions: $h(t) \geq 0, \quad m(t) \geq 0$.

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