In "Modeling and Control of Complex Physical Systems – The Port-Hamiltonian Approach" on p. 273-275 input-state-output port-Hamiltonian systems are examined.
These are of the form $$ \begin{cases} \dot x = [J(x) - R(x)] \frac{\partial H}{\partial x} (x) + g(x)u + k(x)d \\ y = g^T(x) \frac{\partial H}{\partial x}(x) \\ z = k^T(x) \frac{\partial H}{\partial x}(x)\end{cases} $$
Where $J(x) = -J^T(x), R(x) = R^T(x) \geq 0$.
(In the case I'm interested in, $u = y = d = z = 0$.)
The energy-balance equation for such a system is given by $$ H(x(t)) - H(x(0)) = \int_0^t u^T(s) y(s)~\mathrm{d}s - d(t)$$ where $H$ is the system's hamiltonian function and $d(t)$ is a non-negative function capturing dissipative effects.
It is stated that:
The energy of the uncontrolled system (i.e. with $u \equiv 0$) is non-increasing (that is, $H(x(t)) \leq H(x(0))$) and it will actually decrease in the presence of dissipation.
If the energy function is bounded from below, the system will eventually stop at a point of minimum energy.
While the first part of that statement is clear to me, I can't figure out, why the statement about the convergence holds.
The same statement is also found in Putting Energy Back in Control on page 4 of the pdf.