Controlling a Dynamic System - Part 2

Calculus Level 3

This is a follow-up problem to this .

Consider an arbitrary physical system, the behaviour of which is determined by the following differential equation:

$\frac{dx}{dt} = x + u$

Here, $x$ is a time-varying quantity of the system and $u$ is a time-varying input to the system. The system is given a time-varying input, which depends on the variable $x$ , such that the energy of the system always decreases with time . The input is of the form:

$u = \pm x \pm kx^3$

Finding the correct input leads to the answer to the previous version of this problem. The goal of this question is to find the positive number $k$ . It is required that after $t = 5$ seconds, the energy of the system must be $E(5) = 12.5$ units.

Enter your answer as $1000k$

Initial condition: $x(0) = 10$ .

The energy of this arbitrary system is:

$E = \frac{x^2}{2}$

The answer is 3.

1 solution

Steven Chase
Jul 24, 2019

From the last problem:

$u = -x - kx^3$

Therefore:

$\frac{dx}{dt} = x + u = - kx^3 \\ dt = -\frac{1}{k} \, x^{-3} \, dx$

Integrating both sides:

$5 = -\frac{1}{k} \, \int_{10}^5 x^{-3} \, dx$

Crunching out the integral and doing some manipulation yields:

$k = \frac{3}{1000}$

Thanks for the solution. I have posted a couple of follow-up problems on this subject.

https://brilliant.org/problems/converting-differential-to-difference-equation/?ref_id=1573735

https://brilliant.org/problems/controlling-a-dynamic-system-part-3/?ref_id=1573738

Karan Chatrath - 1 year, 10 months ago

Controlling a Dynamic System - Part 2

The answer is 3.

1 solution

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