Suppose I have the following state space system: $$ \dot{x}(t) = Ax(t) + Bu, \quad y(t) = Lx(t), \quad x(0) = x_0 $$ where $A$, $B$ and $L$ are real matrices, $u$ is a constant real vector (so that the system is subjected to a step input at $t=0$), and $L$ is just an identity matrix. In addition, it is also known that all eigenvalues of $A$ are real and negative.

Given this information, can I know (without solving the system in time) if all states of this system will have a monotonic behaviour (i.e. if a given state is increasing it will keep on increasing and vice versa)? in other words, if any state of the system has "overshoot" like behaviour, is there some sufficient condition to detect it?


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