543 research outputs found
Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme
Here we present well-posedness results for first order stochastic
differential inclusions, more precisely for sweeping process with a stochastic
perturbation. These results are provided in combining both deterministic
sweeping process theory and methods concerning the reflection of a Brownian
motion. In addition, we prove convergence results for a Euler scheme,
discretizing theses stochastic differential inclusions.Comment: 30 page
Feedback stabilization and Lyapunov functions
International audienceGiven a locally defined, nondifferentiable but Lipschitz Lyapunov function, we construct a (discontinuous) feedback law which stabilizes the underlying system to any given tolerance. A further result shows that suitable Lyapunov functions of this type exist under mild assumptions. We also establish a robustness property of the feedback relative to measurement error commensurate with the sampling rate of the control im- plementation scheme
Necessary and sufficient condition on global optimality without convexity and second order differentiability
The main goal of this paper is to give a necessary and sufficient condition
of global optimality for unconstrained optimization problems, when the objective
function is not necessarily convex. We use Gâteaux differentiability of the objective
function and its bidual (the latter is known from convex analysis)
Mass Transportation on Sub-Riemannian Manifolds
We study the optimal transport problem in sub-Riemannian manifolds where the
cost function is given by the square of the sub-Riemannian distance. Under
appropriate assumptions, we generalize Brenier-McCann's Theorem proving
existence and uniqueness of the optimal transport map. We show the absolute
continuity property of Wassertein geodesics, and we address the regularity
issue of the optimal map. In particular, we are able to show its approximate
differentiability a.e. in the Heisenberg group (and under some weak assumptions
on the measures the differentiability a.e.), which allows to write a weak form
of the Monge-Amp\`ere equation
The boundary Riemann solver coming from the real vanishing viscosity approximation
We study a family of initial boundary value problems associated to mixed
hyperbolic-parabolic systems:
v^{\epsilon} _t + A (v^{\epsilon}, \epsilon v^{\epsilon}_x ) v^{\epsilon}_x =
\epsilon B (v^{\epsilon} ) v^{\epsilon}_{xx}
The conservative case is, in particular, included in the previous
formulation.
We suppose that the solutions to these problems converge to a
unique limit. Also, it is assumed smallness of the total variation and other
technical hypotheses and it is provided a complete characterization of the
limit.
The most interesting points are the following two.
First, the boundary characteristic case is considered, i.e. one eigenvalue of
can be .
Second, we take into account the possibility that is not invertible. To
deal with this case, we take as hypotheses conditions that were introduced by
Kawashima and Shizuta relying on physically meaningful examples. We also
introduce a new condition of block linear degeneracy. We prove that, if it is
not satisfied, then pathological behaviours may occur.Comment: 84 pages, 6 figures. Text changes in Sections 1 and 3.2.3. Added
Section 3.1.2. Minor changes in other section
Regularity of a kind of marginal functions in Hilbert spaces
We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient
Parallelization of the discrete gradient method of non-smooth optimization and its applications
We investigate parallelization and performance of the discrete gradient method of nonsmooth optimization. This derivative free method is shown to be an effective optimization tool, able to skip many shallow local minima of nonconvex nondifferentiable objective functions. Although this is a sequential iterative method, we were able to parallelize critical steps of the algorithm, and this lead to a significant improvement in performance on multiprocessor computer clusters. We applied this method to a difficult polyatomic clusters problem in computational chemistry, and found this method to outperform other algorithms. <br /
Singular Casimir Elements of the Euler Equation and Equilibrium Points
The problem of the nonequivalence of the sets of equilibrium points and
energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian
formulation of equations describing ideal fluid and plasma dynamics, is
addressed in the context of the Euler equation for an incompressible inviscid
fluid. The problem is traced to a Casimir deficit, where Casimir elements
constitute the center of the Lie-Poisson algebra underlying the Hamiltonian
formulation, and this leads to a study of the symplectic operator defining the
Poisson bracket. The kernel of the symplectic operator, for this typical
example of an infinite-dimensional Hamiltonian system for media in terms of
Eulerian variables, is analyzed. For two-dimensional flows, a rigorously
solvable system is formulated. The nonlinearity of the Euler equation makes the
symplectic operator inhomogeneous on phase space (the function space of the
state variable), and it is seen that this creates a singularity where the
nullity of the symplectic operator (the "dimension" of the center) changes.
Singular Casimir elements stemming from this singularity are unearthed using a
generalization of the functional derivative that occurs in the Poisson bracket
Crime as risk taking
Engagement in criminal activity may be viewed as risk-taking behaviour as it has both benefits and drawbacks that are probabilistic. In two studies, we examined how individuals' risk perceptions can inform our understanding of their intentions to engage in criminal activity. Study 1 measured youths' perceptions of the value and probability of the benefits and drawbacks of engaging in three common crimes (i.e. shoplifting, forgery, and buying illegal drugs), and examined how well these perceptions predicted youths' forecasted engagement in these crimes, controlling for their past engagement. We found that intentions to engage in criminal activity were best predicted by the perceived value of the benefits that may be obtained, irrespective of their probabilities or the drawbacks that may also be incurred. Study 2 specified the benefit and drawback that youth thought about and examined another crime (i.e. drinking and driving). The findings of Study 1 were replicated under these conditions. The present research supports a limited rationality perspective on criminal intentions, and can have implications for crime prevention/intervention strategies
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