Initial rest condition for the linear constant-coefficient differential equations

Question

Suppose that system has the input/output relation as follows

$$\sum_{k=0}^{N}a_k \frac{d^ky(t)}{dt^k} = \sum_{k=0}^{M}b_k \frac{d^kx(t)}{dt^k}$$

Where $a_k, b_k \in \mathbb{R}$.

Obviously we need additional conditions to determine completely the input-output relationship for the system and those auxiliary conditions specify whether the system is LTI and causal or not.

The definition of initial rest condition for a system in general is that if $x(t) = 0$ for $t\lt t_0$ then $y(t) = 0$ for $t\lt t_0$.

First of all is it true that the initial rest condition for the all systems implies $y(t_0) = 0$? Also I have seen expressions like $y({t_0}^+)$ and $y({t_0}^-)$ but I don't know how they relate to $y(t_0)$. Another question is that whether the initial rest condition is necessary and sufficient for the mentioned system to be causal and LTI? What's the rigorous proof, then? I searched on the Internet but didn't found a proof or a clear explanation.

Answer 1

The notion of initial rest and causality follows only for linear systems (time invariant not required).

For proof of Initial rest being sufficient condition for an LTI system to be causal. Please refer to Problem 1.44 of Allen Oppenheim's book on Signal and Systems. Try and solve it yourself (mostly a homework problem, there is also a solution manual just to let you know in case you are stuck), it doesn't require anything extra other than convolution sum.

For $y(t_o^+)$ and $y(t_o^-)$ relating to $y(t_o)$ post the quote you are looking for clarifications on, it depends on multiple context, generally that would be specifying an initial condition of a one sided Fourier transform or one sided sequence.