Problem: 5 - Give a formal definition of a “stable system”. Give a formal definition of an unstable system.
Solution: Consider system f, where input x(t) yields output y(t) = f(x(t)). Consider bounded x(t), i.e. $ \exists M_1 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |x(t)|\leq M_1 $. The system f can be considered stable iff $ \forall x(t), \exists M_2 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |f(x(t))| \leq M_2 $
(ie, The system f is stable iff for all x(t), if x(t) is bounded, then f(x(t)) is bounded as well.)
Consider system f, where input x(t) yields output y(t) = f(x(t)). Consider bounded x(t), i.e. $ \exists M_1 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |x(t)|\leq M_1 $. The system f can be considered unstable iff $ \exists x(t) \text{ s.t. } \nexists M_2 \in \mathbb{R} \text{ s.t. } \forall t \in \mathbb{R}, |f(x(t))| \leq M_2 $
(ie, The system f is stable iff there exists a bounded x(t) for which f(x(t)) is not bounded.)
For an LTI (Linear, Time-Invariant) system f, where $ h(t) = f(\delta(t)) $, the system f is considered stable iff $ \int_{-\infty}^\infty |h[\tau ]| \,d \tau $ is finite and unstable iff the above expression is infinite.
(Note: The above is for continuous time (CT) systems. Discrete time (DT) systems behave in almost exactly the same way; simply substitute all "(t)" with "[n]" (e.g., substitute "x(t)" with "x[n]", etc.) and integration with summation.)
