# “Time-varying” and “nonautonomous” dynamical systems and their Lyapunov analysis

It is possible to distinguish the properties "time-varying" and "nonautonomous" in dynamical systems regarding Lyapunov stability analysis?

Does it make a difference if the system depends explicitly on $t$ or indirectly on $t$ due to a time-varying parameter?

I want to explain the problem in detail:

Let a dynamical system denoted by $\dot x = f$, with state $x$. We say that a dynamical system is nonautonomous if the dynamics $f$ depend on time $t$, i.e. $$\dot x = f(t,x).$$

For instance the systems $$\dot x = - t x^2$$ and $$\dot x = -a(t)x,$$ are nonautonomous. Let $a(t)$ be a bounded time-varying parameter, i.e. $||a(t)||<a^+$ and strictly positive, i.e. $a(t) > 0$.

Particularly, the second example is more likely denoted as a time-varying linear system, but of course it is nonautonomous.

In Lyapunov stability analysis autonomous and nonautonomous systems must be strongly distinguished to make assertions about stability of the system, and the Lyapunov analysis for nonautonomuos systems is much more difficult.

And here for me some questions arise. When i want to analize stability of the second example must i really use the Lyapunov theory for nonautonomous systems?

It follows for the candidate $V = 1/2 x^2$

$$\dot V = -a(t)x^2,$$

which is negative definite. Is the origin really asymptotically stable, as i suppose, or must i take the nonautomous characteristic into account in this case?

I would suppose it makes a difference if a system depends explicitly on $t$ as in the first example or just indirect due to a time-varying parameter, since $t$ approaches infinity, but a parameter does not.

You have the definition of "nonautonomous" incorrect. Please consult authoritative references on control theory such as one by Sastry, Khalil or Slotine and Li

Here is a brief summary of the terms as copied from math stackexchange

A system is time invariant if the system parameters does not depend on time

These systems are represented by:

$$\dot x = f(x,u), \dot x = f(x)$$ or when the system is linear $$\dot x = Ax + Bu, \dot x = Ax$$

A system is time varying if the system parameters does depend on time

These systems are represented by:

$$\dot x = f(x,u,t)$$ or $$\dot x = A(t) x + B(t)u$$

For a RLC circuit, $A(t)$ could represent the matrix containing time varying capacitance, inductance or resistance. Similarly, for a mass-spring-damper system, $A(t)$ could represent time varying damping, friction, and mass. Of course, all real system are time varying albeit on the scale of hours, years, or even millennia.

A (time invariant) system is autonomous if the input $u$ is a function of the state:

These systems are represented by:

$$\dot x = f(x,u(x)) = f(x)$$ or $$\dot x = Ax + Bu(x) = (A-BK)x$$ Supposing that we are using feedback $u = -Kx$. Any state feedback systems are autonomous, because your input $u$ is a function of your state.

And you might have guessed it, a (time invariant) non-autonomous system is when your input is not a function of the state

These systems are represented by:

$$\dot x = f(x,u)$$ or $$\dot x = Ax + Bu$$

For example, $u$ could be the irradiation of the sun hitting a solar panel, where as $x$ encapsulates the states of the solar panel. The solar panel is not going to affect sunshine or the sun for that matters, or the cloud passing the sun.

For your question, you (most likely)* cannot use the Lyapunov function as proposed for your system, namely:

Using $$V(x) = \dfrac{1}{2}x^2$$

to prove stability of origin for

$$\dot x = -a(t)x$$

Because your system has time varying parameters. It is autonomous, and time varying.

What you need to do is to construct a time varying Lyapunov function, and in the process you will encounter when a Lyapunov function () is said to be descrescent, etc. Those are not a part of the classical Lyapunov theory, which deals with time-invariant, autonomous system. Your best reference is the control theory text of Slotine and Li.

• Note: I am not that familiar with time varying Lyapunov functions

There is no difference between the two types that you mention. In fact my problem is understanding why you are able to distinguish them in your mind (honestly I don't understand, sorry for this), since there is no mathematical difference between them.

True that nonautonomous systems can be made autonomous by adding some variables (some equations) but you need to know what you are doing because often adding these equations (it need not be $t'=1$ the type of stability changes).