## Section: Scientific Foundations

### Positive semi-definite functions

Participants : Boris Kalitine, Rachid Chabour.

A well known theorem due to Lyapunov allows us to conclude to the asymptotic stability of an equilibrium: consider a system of differential equations (with f(0) = 0 ), if there exists a positive definite function Vsuch that V· f( x) is negative definite, then we can assert that 0 is an asymptotically stable equilibrium point. The knowledge of such a Lyapunov function is often necessary for the design of a stabilizing feedback but it is quite difficult to find such a function.

A little-known result in Occident, due to Kalitine and Bulgakov, allows us to make use of
*semi-definite* positive functions in the investigation of the stability of an equilibrium. Searching such functions is obviously easier and their use simplifies the design of stabilizing feedback laws. Notice that, on this subject, there exists a great amount of works in the literature
of the countries of Eastern Europe: the original result has been extended to discrete-time systems, to PDE, to non autonomous periodic systems, etc. The team is working to extend these results to non-autonomous systems and stochastic systems.