Lyapunov theory is central to the results produced by MAC team members.
It mainly amounts at looking for Lyapunov functions used as certificate of stability and performance (H2, H∞ or impulse-to-peak) for the closed-loop controlled system in different contexts. For robustness purpose, parameter-dependent Lyapunov functions based on matrix polynomials are produced. This framework may be easily extended to the case of discrete-time periodic uncertain systems by considering periodic Lyapunov functions. Associated with LaSalle principle, piecewise quadratic Lyapunov functions are employed to determine stability domains of systems with non-linearities. Lyapunov-Krasovskii type functionals are determined for the case of systems with time-delays. Last, infinite dimensional Lyapunov functions are created for systems described by Partial Differential Equations (PDEs). Lyapunov theory is employed as well for output-feedback design for linear and non-linear systems where, in a backwards procedure, the controller is derived given a choice of Lyapunov function. In this context, the controller could have either continuous dynamics or mixed discrete-continuous dynamics (hybrid controller).

In parallel to these Lyapunov-based results, MAC team members exploit as well alternative theoretical orientations. In addition to stability, passivity and small-gain properties are studied. Sector conditions are exploited and Lur'e functionals combine non-linearity and parametric uncertainty issues. These results, within a dissipativity theoretical framework, lead to contributions for robustness analysis, performance specification and for control design. Another extension is topological separation theory. As demonstrated on robustness problems and for time-delay systems, it gives new possibilities to handle implicit systems and to reduce conservatism of existing results.

The constructive algorithms produced by MAC team are mainly characterized by efficient numerical optimization tools. One of these is Semi-Definite Programming (SDP) with associated Linear Matrix Inequality (LMI) formalism. More generally the team is involved in exploiting and developing optimization methods for solving the problems derived using the above theories. For problems that cannot have convex LMI formulations, other heuristics based for example, on Bilinear Matrix Inequalities (BMIs), are produced.

This optimization oriented viewpoint is indeed at the origin of new theoretical contributions. For instance, the "slack variables" technique is nothing but a reformulation of the Lagrangian duality allowing to convexify optimization problems and to evaluate conservatism of sufficient conditions. The Generalized Problem of Moments (GPM) is strongly related to this last issue. GPM is an infinite-dimensional linear optimization problem on a convex set of measures, intractable numerically in its full generality. On the other hand, if the support of the measures is contained in a compact basic semi-algebraic set, and the functions involved are polynomials, then one can define a numerical scheme based on a hierarchy of LMIs, which provides a monotone non-decreasing sequence of lower bounds that converges to the optimal value of the GPM; sometimes finite convergence may even occur.

Optimal control theory is also used for the proof of some controllability properties. Inverse problems can be seen as a dual question as soon as the point is to determine parameters of a PDE from measurement of the solution. For nonlinear systems, optimal control theory is also important through their connections to hybrid systems (for the design of robust nearly-optimal stabilizing control laws) or to moments approach (for the numerical computation of optimal controller via occupation measures).