We continued our work with D. Goeleven, from the University of la Réunion on systems of the type for all vK, x(t)K (K being a nonempty closed convex set). Necessary conditions for the asymptotic stability have been obtained in , where the notion of Brouwer degree of a function is used. The difficulty here is that the function is nonsmooth. The results are applied to the asymptotic stabilization of a controlled variational inequality. They show that in general the controllability of the unconstrained system, is neither sufficient nor necessary to assure the existence of a constant feedback which guarantee asymptotic stability.

The LaSalle invariance principle has also been studied for the case of evolution variational inequalities, see Sect. .

We recall that the absolute stability problem consists of studying the stability of a system made of the negative feedback interconnection of a dissipative system with a nonlinear characteristic. Last year in we extended absolute stability to a feedback branch containing a maximal monotone operator. These results have been used subsequently in to design stable observers (open-loop ), and feedback controllers using observed states (closed-loop ). There is an additional difficulty in the sense that we now consider inclusions which are non-autonomous. All of these works use well-posedness results for monotone diferential inclusions à la Brézis.

As a sequel to our previous works , we clarified in the design and the role of the transition phases (stabilisation on the constraint surface or detachment from this surface) in the closed-loop stability, extending the so-called passivity-based control design to unilaterally constrained systems. In , the robustness of these family of controllers has been tested with numerical simulations.

The mathematical analysis of nonsmooth Lagrangian dynamical systems requires mathematical tools which are unusual in control theory: velocities with locally bounded variations, measure accelerations, measure differential inclusions, to name a few. The control theory for such dynamical systems is just beginning to appear, and even the basic Lyapunov stability theory still needs to be stated. In , we propose a Lyapunov stability theorem for dynamical systems with state discontinuities. Based on this theorem, we are then able to propose a Lagrange-Dirichlet theorem for nonsmooth Lagrangian dynamical systems that can be applied through Potential Shaping to the regulation of the position and force of a robotic manipulator . Note that though we have been able to derive a Lyapunov stability analysis for nonsmooth dynamical systems very similar to what appear in the smooth case, the derivation of a theorem equivalent to the LaSalle's invariance theorem is far from being straighforward. A key condition for the statement of Lasalle's invariance theorem is the continuity of the trajectories of the systems with respect to initial conditions. Nonsmooth Lagrangian dynamical systems generally do not present such a continuity, but they do in many specific cases. So we propose in a version of LaSalle's invariance theorem for time-invariant flows that are countinuous with respect to initial conditions.

First experiments of standing with different task function control laws have been undertaken, but the results in tracking reference trajectories revealed deficiencies in the velocity observer and the friction compensation. Then a breakage of the mechanical structure (left hip) stopped experimentations for 4 months; as a result, our work on improvement of the trajectory tracking resumed by the end of the year.

The modeling and simulation software for humanoid robots and electro-stimulated paraplegic people, developed the team in C/C++/Scilab/Maple, is used at Inria Rhône-Alpes, LMS in Poitiers and LIRM in Montpellier. It is soon to reach its first official release, with more realistic models of the sensors of the Bip robot, and a complete dynamical model of the electro-stimulated muscles of paraplegic people.

Work done with Stäubli SCA, Faverges (Luc Joly).

Trajectory generation in industry is usually based on simple laws like bang-bang profile. Such a profile is not adapted to industrial problems because it makes it difficult to search the best trajectory for a given task. Accordingly, we formulate the problem of trajectory generation as a constrained optimization problem. We model the robot (geometry, kinematics and dynamics) and the physical constraints (on actuators, temperature, etc.) in a traditional way. For a numerical resolution, the profile is discretized by cubic splines and the constrained optimization problem is solved by a Sequential Quadratic Programming solver. Our algorithm is robust to the initial conditions, and it can be implemented as an off-line optimization tool; see .

Work done with A. Daniilidis (Autonomous Univ. of Barcelona), a former post-doctoral fellow at Inria Rhône-Alpes.

A complementarity model such as () is actually an instance of more general differential inclusions, where the speed is constrained to lie in some set T. Equivalence of a few such models was known before under fairly ad hoc assumptions . Using exclusively tools from standard convex and nonsmooth analysis, we establish in the same equivalence with several other models, under much simpler assumptions; this work in particular reveals the importance of the concept of slow solution .

An impact is said to be multiple when several impacts occur at the same time on the system. The multiple impact mappings which have already been proposed in the literature are not satisfactory, because they often entail post-impact velocities that obviously disagree with experiments, and/or are not tractable from the numerical point of view. Based on previous works in Bipop , , the goal of the PhD thesis D.E. Taha in collaboration with Schneider Electric is to extend the multiple impact law to finite-dimensional systems such as circuit breakers and to exhibit the limit of validity of the approach (assumption of quasi-rigid solids).

Work done with A. Daniilidis (Autonomous Univ. of Barcelona), a former post-doctoral fellow at Inria Rhône-Alpes.

Important nonsmooth functions are lower-C¹ and lower-C² functions (a lower-C^k function is the supremum of C^k functions). We have defined and studied in a class of intermediate functions, called lower-; they play a role analogous to Hölder functions in standard analysis.

Work done in collaboration with Gérard Cornuéjols (Carnegie Mellon) and funded by the Inria new investigation grant ODW, extending over 2003-04.

One of the crucial techniques in combinatorial optimization is separation: given a closed convex set P and a point , find a hyperplane separating from P ``best''. Traditional studies of this problem are limited to a polyhedral P and rely heavily on LP theory. In (submitted to Mathematical Programming), we consider this question from the higher point of view of convex analysis, confirming the importance of the reverse polar of a set Q:

It is the set of directions separating Q from 0 strictly, and is at the same time a closed set, which considerably eases theory and computations. We show how this new perspective can be applied to disjunctive cuts . This synthetic point of view helps understanding better the issue, suggests an appropriate definition of ``best'' separations, and opens the way to generalizations (say to SDP relaxations).

Work done with S. Miller (Univ. of Cal. at San Diego), a former post-doctoral fellow at Inria Rhône-Alpes.

The concept of smooth restriction of a nonsmooth function f to a smooth manifold was exhibited in ; it is fundamental to design fast algorithms minimizing f. Making use of its formalization , we analyze in (submitted to Mathematical Programming) fruitful links with Riemannian geometry , SQP theory , proximal mapping . Our study reveals the importance of the existence of the so-called U-gradient (essentially: the gradient of the smooth part of f), without which developing the smooth part of f to second order is hardly conceivable.

Recall (see for example) that SDP relaxations of a combinatorial problem replace the variable x{-1, + 1}ⁿ by a symmetric n×n matrix X (standing for ) of rank 1 and whose diagonal entries are all 1 (then they relax the rank-one constraint to ).

In we note that the set of such matrices can also be described by the constraits and . Then we propose to dualize this latter single quadratic constraint, say with a multiplier . For = 0, the optimal value of the Lagrangian is the SDP bound, while the duality gap is closed when -. On the other hand, the Lagrangian problem for >0 is a so-called SDLS problem, which is substantially easier than a standard linear SDP: . This opens the way to potentially more efficient relaxations than SDP. Besides, an interesting class of nonconvex problems with no duality gap is thus revealed.

Work done in collaboration with F. Vanderbeck (Univ. Bordeaux 1) and K.C. Kiwiel (Systems Research Institute, Warsaw) and funded by the Inria new investigation grant ODW, extending over 2003-04; see Numopt Activity Report 2003, §6.2).

For technical reasons the team from Geneva had to withdraw from this project. Our continued collaboration with the team from Bordeaux confirmed last year's results: compared with the traditional Dantzig-Wolfe algorithm, the bundle approach is always competitive and sometimes drastiscally faster. This, however, holds in terms of numbers of calls to the oracle. On the other hand, we observed strange behaviours: an NP oracle may take considerably more CPU time when called by bundle. It is therefore crucial for bundle to accept inaccurate oracles, thereby reducing computing times (see Numopt Activity Report 2003, §6.3). Accordingly, we implemented with K.C. Kiwiel his version of bundle with noise .

This work was also the occasion to revisit stabilization of Dantzig-Wolfe (see among others), relating it with the Moreau-Yosida regularization in the dual space. These results are published in , submitted to Mathematical Programming.