## Related questions with answers

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

Verified$\textbf{A moon lander}$ 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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