On some aspects of polynomial dynamical systems

The aim of this work is to study exact algebraic criteria local/global observability ([HK77], [Ino77]) for polynomial dynamical system by means of algebraic geometry and computational commutative algebra in the vein of [SR76], [Son79a], [Son79b], [Bai80], [Bai81], [Bar95], [Bar99], [Nes98], [Tib04], [KO13], [Bar16]. A key point in this topic is to work with polynomials with real coefficients and their real roots instead of their complex roots, as it is usually the case ([CLO15], [KR00]). A central concept is then the real radical of an ideal [BN93], [Neu98], [LLM+13], along with the Krivine- Dubois-Risler real nullstellensatz for polynomial rings [Kri64], [Dub70], [Ris70], [BCR98]. Underestimating this point leads to incorrect results (see, e.g. [Bar16] remark on [KO13]). This thesis is therefore devoted to set the necessary algebraic tools in the right context and level of generality (i.e. real algebra and real algebraic geometry) for applications to our dynamical systems and to further develop their exploit in this context. The first two chapters set the algebraic and algebraic geometry preliminaries. The third chapter is devoted to the applications of the previous algebraic concepts to the study of the ob- servability of polynomial dynamical systems. In the last chapter an approach to the construction of Lyapunov funtions to prove stability in estimation problems is presented.