On the stability of an LTI circuit

[mjtsquared]

An LTI circuit such as one composed of resistors, capacitors, and inductors, in general is a stable LTI system, i.e. its impulse response is one that decays over time. I have no problem with that, as it speaks for itself through laws of energy conservation, but I want to see this from a mathematical standpoint.

Following from this assumption, the transfer function of the system must have poles on the left-half side of the complex plane i.e. the real parts of the potentially complex roots of the denominator polynomial are negative. I know about the Routh-Hurwitz stability criterion, and have used it on many examples which do pass the criterion, but I still can't find any generality. If it's something that always allows this to happen, it must be something with Kirchhoff's Laws. What do you think?

Good day!

 

[FactChecker]

If an impulse was fed in and the energy in the system grew exponentially without bound, that would be generating energy from nothing. That rules out divergent behavior. Also, any resistance at all would bleed off energy. That rules out a stable continuous oscillation.