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...

We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.

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...

Let $M$ be a smooth connected complete manifold of dimension $n$, and $\Delta $ be a smooth nonholonomic distribution of rank $m\le n$ on $M$. We prove that if there exists a smooth Riemannian metric on1for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $\Delta $ on $M$. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of...

Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka [ (2001) 21–40], is due to Li and Nirenberg...

An optimal control problem is studied, in which the state is required to remain in a compact set $S$. A control feedback law is constructed which, for given $\epsilon \>0$, produces $\epsilon $-optimal trajectories that satisfy the state constraint universally with respect to all initial conditions in $S$. The construction relies upon a constraint removal technique which utilizes geometric properties of inner approximations of $S$ and a related trajectory tracking result. The control feedback is shown to possess a robustness...

An optimal control problem is studied, in which the state is required
to remain in a
compact set . A control feedback law is constructed which, for
given ε > 0, produces -optimal trajectories that satisfy the
state constraint universally with respect to all initial conditions
in .
The construction relies upon a constraint removal technique which
utilizes geometric properties of inner approximations of and a
related trajectory tracking result.
The control feedback is shown to possess a robustness...

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