We prove that quasiconvex functions always admit descent trajectories bypassing all nonminimizing critical points.

This work models and analyses the dynamics of a general spring-mass-damper system that is in frictional contact with its support, taking into account frictional heat generation and a reactive obstacle. Friction, heat generation and contact are modelled with subdifferentials of, possibly non-convex, potential functions. The model consists of a non-linear system of first-order differential inclusions for the position, velocity and temperature of the mass. The existence of a global solution is established and additional assumptions yield its uniqueness. Nine examples of conditions arising in applications, for which the analysis results are valid, are presented.

The existence of solutions to a homogeneous Dirichlet problem for a p-Laplacian differential inclusion is studied via a fixed-point type theorem concerning operator inclusions in Banach spaces. Some meaningful special cases are then worked out.

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the underlying mathematical framework, and we explain how recent results by J.F. Edmond and L. Thibault on the sweeping process by uniformly prox-regular sets can be adapted to handle this situation in terms of well-posedness. We propose a numerical scheme for this contact dynamics model, based on a prediction-correction algorithm. Numerical illustrations are finally presented and discussed.

In this work we study the optimal control problem for a class of nonlinear time-delaysystems via paratingent equation with delayed argument. We use an equivalence theorembetween solutions of differential inclusions with time-delay and solutions of paratingentequations with delayed argument. We study the problem of optimal control which minimizes acertain cost function. To show the existence of optimal control, we use the maintopological properties of the set solutions of paratingent equation with delayedargument

We consider a class of variationalproblems for differential inclusions, related to thecontrol of wild fires. The area burned by the fire at time t> 0is modelled as the reachable set fora differential inclusion $\dot x$ ∈F(x), starting froman initial set R 0. To block the fire, a barrier can be constructedprogressively in time. For each t> 0, the portion of the wall constructedwithin time t is described by a rectifiable setγ(t) ⊂ $\mathbb{R}^2$ . In this paperwe show that the searchfor blocking strategies and for optimal strategies can be reduced toa problem involving one single admissible rectifiable set Γ⊂ $\mathbb{R}^2$ ,rather than the multifunction t $\mapsto$ γ(t) ⊂ $\mathbb{R}^2$ .Relying on this result, we then developa numerical algorithm for the computation ofoptimal strategies, minimizing the total area burned by the fire.

Let $\dot{x}=f(x,u)$ be a general control system; the existence of a smooth control-Lyapunov function does not imply the existence of a continuous stabilizing feedback. However, we show that it allows us to design a stabilizing feedback in the Krasovskii (or Filippov) sense. Moreover, we recall a definition of a control-Lyapunov function in the case of a nonsmooth function; it is based on Clarke's generalized gradient. Finally, with an inedite proof we prove that the existence of this type of control-Lyapunov function is equivalent to the existence of a classical control-Lyapunov function. This property leads to a generalization of a result on the systems with integrator.