• 04 October : PhD Defence of Tung PHAN THANH, salle de Conférences.
  Title:   Contributions to the Moment-SOS Approach in Glo! bal Polynomial Optimization
  Jury :    

  Monique LAURENT, Pham DINH TAO, Léo LIBERTI, Jean-Baptiste HIRIART-URRUTY, Jean-Bernard LASSERRE

  Abstract:    

  Polynomial Optimization is concerned with optimization problems of the form (P) :  f* = { f(x) with x in set K}, where K is a basic semi-algebraic set in Rn defined by K={x in Rn such as gj(x) less or equal 0}; and f is a real polynomial of n variables x = (x1, x2, ..., xn).
  In this thesis we are interested in problems (P) where symmetries and/or structured sparsity are not easy to detect or to exploit, and where only a few (or even no) semidefinite relaxations of the moment-SOS app! roach can be implemented. And the issue we investigate is: How! can the ! moment-SOS methodology be still used to help solve such problem (P)? We provide two applications of the moment-SOS approach to help solve (P) in two different contexts.
  
  * In a first contribution we consider MINLP problems on a box B = [xL, xU] of Rn and propose a moment-SOS approach to construct polynomial convex underestimators for the objective function f (if non convex) and for -gj if in the constraint gj(x) less or equal 0, the polynomial gj is not concave. We work in the context where one wishes to find a convex underestimator of a non-convex polynomial f of a few variables on a box B of Rn. The novelty with previous works on this topic is that we want to compute a polynomial convex underestimator p of f that minimizes the important tightness criterion which is the L1 norm of (f-h) on B, over all convex polynomials h of degree d _fixed.
  
  * In a second contribution we propose an algorithm that also uses an optimal solution of a semidefinite relaxati! on in the moment-SOS hierarchy (in fact a slight modification) to provide a feasible solution for the initial optimization problem but with no rounding procedure. In the present context, we treat the first variable x1 of x = (x1, x2, ...., xn) as a parameter in some bounded  interval Y of R. Notice that f*=min { J(y) : y in Y} where J is the function J(y) := inf {f(x) : x in K ; x1=y}. That is one has reduced the original n-dimensional optimization problem (P) to an equivalent one-dimensional optimization problem on an interval. But of course determining the optimal value function J is even more complicated than (P) as one has to determine a function (instead of a point in Rn), an infinite-dimensional problem. But the idea is to approximate J(y) on Y by a univariate polynomial p(y) with the degree d and fortunately, computing such a univariate polynomial is possible via solving a semidefinite relaxation associated with the parameter optimization problem. The degree d of p(y! ) is related to the size of this semidefinite relaxation.

• 03 December : PhD Defence of Jean-François TREGOUET, salle de Conférences.
  Title:  Synthèse de correcteurs robustes périodiques à mémoire et application au contrôle d'attitude de satellites par roues à réaction et magnéto-coupleurs
  Jury :    

  Olivier SENAME, Marco, LOVERA, Jamal DAAFOUZ, Hélène PIET LAHANIER, Christelle PITTET, Dimitri PEAUCELLE, Denis ARZELIER, Daniel ALAZARD. 

  Abstract:    

  This manuscript reviews contributions to the development of systematic methods for analysis and control of periodic uncertain systems. An important part of this thesis is also dedicated to the design of attitude co! ntrol systems for satellites whose dynamics is naturally repre! sented as! a periodic model subject to uncertainties.
  
  The first part is devoted to the developpement of a unifying presentation of the analysis and synthesis results of periodic, uncertain and discrete-time models via methods relying on linear matrix inequalities (LMI) and based on Lyapunov theory. Subsequently, the focus is on a new class of periodic control laws with memory for which the control input is constructed using history of the states of the system kept in memory. Numerical experiments show that these new degrees of freedom can outperformed the existing results.
  
  The second part deals with periodic and robustness aspects of attitude control of a satellite using magnetorquers. These actuators use the geomagnetic field that varies periodically along the orbital trajectory. Different control strategies are implemented and compared with one another with the constant concern of taking the main limitations of the actuators into account. This approach leads t! o a new control law regulating the momentum of the reaction wheels without disturbing attitude control for which the control effort is shared by all actuators.