Convergence with increasing Lyapunov function

Question

Given a (autonomous) dynamical system, one can prove instability of a point via the Lyapunov method, by simply finding a Lyapunov function that increases in a neighbourhood of the point. This ensures a stronger property than instability. It ensures the existence of a small enough neighbourhood, such that all orbits starting inside exit the neighbourhood at some time (instability only requires the existence of one such orbit, quite weaker).

My question is the following: does this stronger property allow that the system might converge (or approach) to the point in question? In other words, can a dynamical system converge (or approach) to a point admitting a neighbourhood with (strictly) increasing Lyapunov function? Thanks for any advice or reference.

EDIT: So far we proved that this cannot happen for continuous time systems. Is this also true for discrete time systems? Since the answer is in general settled to a no (continuous systems offer a counterexample as per the answer below) I will raise as a separate question the discrete case : Chetaev theorem for discrete time.

Answer 1

The answer to this question depends on the flavour of theorem you wish to invoke. For instance, here is a theorem from a nonlinear control text one could use,

Theorem 5.29 Instability Theorem. The equilibrium point $0$ is unstable at time $t_0$ if there exists a decrescent function $V: \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}$ such that

1. $\dot V(x,t)$ is locally positive definite, and
2. $V(0, t) = 0$ and there exists points $x$ arbitrarily close to $0$ such that $v(x, t_0) > 0$

Indeed, this flavour only demands that there exists a one-dimensional unstable manifold passing through the origin (condition 2) since the positive definiteness is only assured on some set of points arbitrarily close to the origin. This does not preclude stability towards that set, for instance, including the origin along some set of measure zero. However, if you ask $V$ be locally positive definite except at $x = 0$, then that variant does not support convergence to the origin since all the points near the origin are expelled. The text I follow makes this comment:

The instability theorems have the same flavor: They insist on $\dot V$ being an l.p.d.f. so as to have a mechanism for the increase of $V$. However, since we do not need to guarantee that every initial condition close to the origin is repelled from the origin, we do not need to assume that $V$ is an l.p.d.f.

So it boils down to what theorem you are choosing to use. In [2] there is further detailed comment to this effect for a very similarly stated theorem (although, not as precise as Sastry or Khalil were, so I will not quote the theorem).

It is interesting to note that the function $V(\cdot)$ in Theorem 3.12 can be positive as well as negative in $D$. However, $V(·)$ is required to be positive for some points $x$ arbitrarily close to the origin of the nonlinear dynamical system. A more restrictive version of Theorem 3.12 is the case where $V : D \to R$ is positive definite for all $x\in B_\varepsilon(0)$. In this case, the zero solution $x(t) ≡ 0$ to (3.1) is completely unstable in the sense that there exists $\varepsilon > 0$ such that every trajectory starting in $B_\varepsilon(0)$, other than the trivial trajectory, eventually leaves $B_\varepsilon(0)$.

The reason for this of course derives from the proof of instability. In proving Theorem 5.29, within any ball one picks $x(0)$ so that $V(0) > 0$. If it is the case that $V(0) > 0$ in an open neighbourhood of $0$ excluding $0$, then the proof can be applied to show all trajectories exit the neighbourhood.

[1]: S. Sastry. Nonlinear Systems: Analysis, Stability and Control. 1999.

[2]: W. M. Haddad and V. S. Chellaboina. Nonlinear Dynamical Systems and Control. 2008.