Controlling a Dynamic System

Calculus Level 3

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 goal of this question is to ensure that 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 . Which of the options ensures this requirement?

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

The energy of this arbitrary system is:

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

Note: $k$ is a positive real number.

Also try: this problem

u = x - kx^3

u = x + kx^3

u = -x - kx^3

u = -x + kx^3

1 solution

Karan Chatrath
Jul 24, 2019

Consider the energy of the system:

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

Differentiating with respect to time:

$\dot{E} = \dot{x}{x}$

Substituting $\dot{x} = x+u$ :

$\dot{E} = x(x+u)$

Now, energy must always decrease. In other words, its time-derivative must always be negative. Among the options available, the option that meets this requirement is:

$\boxed{u = -x - kx^3}$

This is because:

$\dot{E} = x(x+u) = x(x - x - kx^3) = -kx^4$

Which is always negative

Nice problem! That's exactly how I did it; the rate of change of energy must be negative, so the only fitting solution is C.

Krishna Karthik - 1 year, 1 month ago

Thanks. Glad you liked it

Karan Chatrath - 1 year, 1 month ago

Controlling a Dynamic System

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