Constructing and exploring wells of energy landscapes

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features ``wells of duration $T$ and altitude $y$''. We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) which have been developed during the last two decades; we therefore propose a minimal appendix exposing them at the end of this paper to bridge the possible gap.Comment: 28 pages, submitted to J. Math. Phys -

The human relations aspects of inaugurating and implementing a program of industrial security

Thesis (M.B.A.)--Boston Universit

Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Systems of first-order partial differential inclusions -- solutions of which are feedbacks governing viable trajectories of control systems -- are derived. A variational principle and an existence theorem of a (single-valued contingent) solution to such partial differential inclusions are stated. To prove such theorems, tools of set-valued analysis and tricks taken from viability theory are surveyed. This paper is the text of a plenary conference to the World Congress on Nonlinear Analysis held at Tampa, Florida, August 19-26, 1992

Hyperbolic Systems of Partial Differential Inclusions

This paper is devoted to the study of first-order hyperbolic systems of partial differential inclusions which are in particular motivated by several problems of control theory, such as tracking problems. The existence of contingent single-valued solutions is proved for a certain class of such systems. Several comparison and localization results (which replace uniqueness results in the case of hyperbolic systems of partial differential equations) allow to derive useful informations on the solutions of these problems

Dynamic Regulation of Controlled Systems, Inertia Principle and Heavy Viable Solutions

Existence of viable (controlled invariant) solutions of a control problem regulated by absolutely continuous open loop controls is proved by using the concept of viability kernels of closed subsets (largest closed controlled invariant subsets). This is needed to provide dynamical feedbacks, i.e., differential equations governing the evolution of viable controls. Among such differential equations, the differential equation providing heavy solutions (in the sense of heavy trends), i.e., governing the evolution of controls with minimal velocity is singled out. Among possible applications, these results are used to define global contingent subsets of the contingent cones which allow to prove the convergence of a modified version of the structure algorithm to a closed viability domain of any closed subset