Dynamic Optimization : Resource Stock as a Sink
"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?"


No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative.
Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: if "we don't care" about natural resources, their "price" is zero.

To see this from another route, for the standard CRRA utility function

$$u(c) = \frac {c^{1-\theta}-1}{1-\theta}$$

we would get

$$\frac {\dot c}{c} = \frac{1}{\theta}[A-\rho] - \frac{1}{\theta}\frac{\mu}{\lambda}\gamma A$$

So if $\mu &lt;0$ we would obtain a higher consumption growth rate, which doesn't sound very plausible if by  $\mu &lt;0$ we would want to express that "we care" about natural resources and so we would lower capital accumulation that hurts them, and consequently lower the consumption growth rate.