17 research outputs found
Stabilizability in optimal control
We extend the well known concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also associated costs; in particular, we introduce the notions of Sample and Euler stabilizability to a closed target set C with W-regulated cost, which roughly means that we require the existence of a stabilizing feedback such that all the corresponding sampling and Euler solutions have finite costs, bounded above by a continuous, state-dependent function W, divided by some positive constant c. We prove that the existence of a special Control Lyapunov Function W, called c-Minimum Restraint function, c-MRF, implies Sample and Euler stabilizability to C with W-regulated cost, so extending [Motta, Rampazzo 2013], [Lai, Motta, Rampazzo, 2016], where the existence of a c-MRF was only shown to yield global asymptotic controllability to C with W-regulated cost
Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization
The paper is concerned with patchy vector fields, a class of discontinuous,
piecewise smooth vector fields that were introduced in AB to study feedback
stabilization problems. We prove the stability of the corresponding solution
set w.r.t. a wide class of impulsive perturbations. These results yield the
robusteness of patchy feedback controls in the presence of measurement errors
and external disturbances.Comment: 22 page
On feedback stabilization of linear switched systems via switching signal control
Motivated by recent applications in control theory, we study the feedback
stabilizability of switched systems, where one is allowed to chose the
switching signal as a function of in order to stabilize the system. We
propose new algorithms and analyze several mathematical features of the problem
which were unnoticed up to now, to our knowledge. We prove complexity results,
(in-)equivalence between various notions of stabilizability, existence of
Lyapunov functions, and provide a case study for a paradigmatic example
introduced by Stanford and Urbano.Comment: 19 pages, 3 figure
On continuation and convex Lyapunov functions
Given any two continuous dynamical systems on Euclidean space such that the
origin is globally asymptotically stable and assume that both systems come
equipped with -- possibly different -- convex smooth Lyapunov functions
asserting the origin is indeed globally asymptotically stable. We show that
this implies those two dynamical systems are homotopic through qualitatively
equivalent dynamical systems. It turns out that relaxing the assumption on the
origin to any compact convex set or relaxing the convexity assumption to
geodesic convexity does not alter the conclusion. Imposing the same convexity
assumptions on control Lyapunov functions leads to a Hautus-like
stabilizability test. These results ought to find applications in optimal
control and reinforcement learning.Comment: 16 pages, comments are welcome. V2: fixed 1 typ
On the construction of nearly time optimal continuous feedback laws around switching manifolds
In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known
Singularities of viscosity solutions and the stabilization problem in the plane
International audienceWe study the general problem of globally asymp- totically controllable affine systems in the plane. As preliminaries we present some results of independent interest. We study the regularity of some sets related to semiconcave viscosity supersolutions of Hamilton-Jacobi-Bellman equations. Then we deduce a construction of stabilizing feedbacks in the plane
On the stabilization problem for nonholonomic distributions
Let be a smooth connected and complete manifold of dimension , and
be a smooth nonholonomic distribution of rank on . We
prove that, if there exists a smooth Riemannian metric on for which no
nontrivial singular path is minimizing, then there exists a smooth repulsive
stabilizing section of on . 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 nonsmooth analysis, of an optimal control problem of Bolza type, for
which we prove that the corresponding value function is semiconcave and is a
viscosity solution of a Hamilton-Jacobi equation, and establish fine properties
of optimal trajectories.Comment: accept\'e pour publication dans J. Eur. Math. Soc. (2007), \`a
para\^itre, 29 page