The study of linear systems [BOU 06] is fundamental for the proper understanding of the concepts of stability and the design of linear controllers. Let us recall that linear systems are of the form:

for continuous-time systems and:

for discrete-time systems.

A linear system is *stable* (also called *asymptotically stable* in the literature) if, after a sufficiently long period of time, the state no longer depends on the initial conditions, no matter what they are. This means (see Exercises 3.1 and 3.2) that:

In this expression, we can see the concept of *matrix exponential*. The exponential of a square matrix M of dimension *n* can be defined through its integer series development:

where **I**_{n} is the identity matrix of dimension *n*. It is clear that *e*^{M} is of the same dimension as M. Here are some of the important properties concerning the exponentials of matrices. If 0_{n} is the zero matrix of *n* × *n* and if M and N are two matrices *n* × *n*, then:

CRITERION OF STABILITY.– There is a ...

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