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## Section: New Results

### Spectral abscissa characterization for Time-delay systems

Participants : Islam Boussaada, Silviu-Iulian Niculescu, Sami Tliba [Université Paris Sud] , Thomas Vyhlidal [Czech technical university in Prague] , Karim Trabelsi [IPSA] .

It is well known that the spectral abscissa of a given dynamical is nothing but the corresponding solutions' exponential decay. The analytical characterization of the spectral abscissa for infinite dimensional dynamical systems is an old problem which is still nowadays a question of ongoing interest due to its links with stability problems. We produced several works in this topic dealing with reduced order retarded Time-delay systems and emphasized a property that we call multiplicity induced-dominancy. In the paper [13], the interest of using time-delay in the controller design as a control parameter is underlined and the way to assign a dominant spectral value is demonstrated. As a matter of fact, it is shown that the multiplicity of given spectral value may reach the degree of the corresponding quasipolynomial. Furthermore, when this holds, then using a particular factorization of the quasipolynomial, such a multiple spectral value is shown to be the corresponding spectral abscissa. A generalization of such a result to generic second order retarded equation with a single delay is established in [12]. More precisely, a parametric characterization of the spectral abscissa is established using the principle argument theorem. Furthermore, in the work [45], the potential applicability of such a parametric characterization in controller design in concrete application is demonstrated. As a matter of fact, a third order retarded system modeling the dynamics of Mach number in a wind tunnel is considered and a delayed controller design based on the spectral abscissa assignment is proposed.