Research Interests

• Computer algebra algorithms 
• Complexity theory 
• Effective real algebraic geometry 
• Algebraic aspects of automatic control

The computation of the L^infinity norm of a dynamical system is well motivated by problems of control theory and robust and optimal control, which requires in some cases the computation of this value with respect to given parameters. This means, for linear dynamical systems depending on real parameters. This norm was largely studied using different methods, where the most classical ones are the numerical approximation methods that are essentially iterative, consisting on computing eigenvalues of given matrices at each iteration. 

I am working on computing this norm through a symbolic approach, consisting of studying curves and surfaces. In fact, this approach show at least the following advantages against the numerical approach: 

• Symbolic methods are well certified, i.e., compute with a finite number of steps and for a given input, a result without ambiguity for the user. Whereas an approximative numerical computation may show numerical errors that may, at some point, lead to an instability and an iteration interruption.
•  Symbolic methods are well adapted to problems depending on real parameters, unlike numerical methods that face major difficulties in presence of parameters. 

But it is worthwhile mentioning that approximative numerical methods stay more advantageous when it comes to the speed of execution and the capacity to treat problems and examples of higher sizes, unlike symbolic methods where the efficiency stays and important factor to deal with when constructing a symbolic approach.

In my latest research work, during my Ph.D., the problem of computing the L^infinity norm of finite dimensional linear systems was reduced to the computation of the L^infinity norm of a rational matrix, which amounts to computing its maximal singular value. This was then reduced to the study of the maximal ordinate of real critical points of a curve, difined by the real solutions of a bivariate polynomial system and computed through different symbolic approaches implemented on Maple. 

The computation was then generalised to the parametric case. 

Currently, I am working on optimizing the constructed algorithms for the non-parametric case by trying to reduce their worst-case bit complexity. I am also working on finilising the constructed approach for the parametric case In order to add the codes to a specified library on Maple. 

In my future research work, I will dive deeper in the algebraic aspects of automatic control.