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I have a problem with one of my study questions for an oral exam:

The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov function for controlling the position and velocity of the system. Consider a damped single degree-of-freedom system, $m\ddot{x}+c\dot{x}+kx=0$, where $m$ is the mass, $c$ is the velocity-proportional damping and $k$ is the stiffness. A candidate Lyapunov function is the Hamiltonian $V=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2$. What are the reasons for leaving out the dissipative energy term when writing the Lyapunov function?

The only thing what comes into my mind for this question is, that a dissipative energy term in the Lyapunov function would have a "-" sign and the Lyapunov function would thus not be positive definite anymore. Is that correct?

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2 Answers 2

I am not quite sure what "dissipative energy term" means, but I do know that you can't add anything proportional to $\dot{x}$. To see why, just take a point close to the $(x, \dot{x})=(0,0)$ point. In a neighbourhood of this point the $\dot{x}$ term will dominate over $\dot{x}^2$ and either the point $(0,\epsilon)$ or $(0,-\epsilon)$ would give a negative value.

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1) In the presence of friction, Lagrange equation gets modified

$$\tag{1} \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}~=~ -\frac{\partial{\cal F} }{\partial \dot{x}}$$

by the Rayleigh dissipation function

$$\tag{2} {\cal F}~: =~ \frac{1}{2} c\dot{x}^2 ~\geq ~0 . $$

Here the Lagrangian is

$$\tag{3} L~:=~T-V, \qquad T~:=~\frac{1}{2} m\dot{x}^2~\geq ~0, \qquad V~:=~\frac{1}{2} kx^2~\geq ~0. $$

It is not possible to write a velocity-dependent potential for the friction force, and a Lagrangian (or Hamiltonian) description of the damped oscillator must be modified a la (1) to accommodate the friction term.

2) The energy function

$$\tag{4} h(x,\dot{x})~:=~ \dot{x} \frac{\partial L}{\partial \dot{x}}-L ~=~T+V~\geq ~0 $$

is precisely the mechanical energy of the system.

One may show that the energy dissipation rate is given by the Rayleigh dissipation function

$$\tag{5} \frac{dh}{dt}~=~-2{\cal F} ~\stackrel{(2)}{\leq} ~0. $$

The positive semi definite (4) of $h$, and the negative semi-definite (5) of the time derivative $\frac{dh}{dt}$ are some of the conditions that one usually demand of a Lyapunov function, and it is not hard to see that the mechanical energy $h$ is in fact a Lyapunov function for the damped oscillator.

On the other hand, it is unclear how to include ${\cal F}$ in the Lyapunov function, for reasons explained above.

References:

  1. Herbert Goldstein, Classical Mechanics, Chapter 1 and 2.
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