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 -

Smooth and Heavy Solutions to Control Problems

We introduce the concept of viability domain of a set-valued map, which we study and use for providing the existence of smooth solutions to differential inclusions. We then define and study the concept of heavy viable trajectories of a controlled system with feedbacks. Viable trajectories are trajectories satisfying at each instant given constraints on the state. The controls regulating viable trajectories evolve according a set-valued feedback map. Heavy viable trajectories are the ones which are associated to the controls in the feedback map whose velocity has at each instant the minimal norm. We construct the differential equation governing the evolution of the controls associated to heavy viable trajectories and we state their existence

Slow and Heavy Viable Trajectories of Controlled Problems. Part 1. Smooth Viability Domains

We define slow and heavy viable trajectories of differential inclusions and controlled problems. Slow trajectories minimize at each time the norm of the velocity of the state (or the control) and heavy trajectories the norm of the acceleration of the state (or the velocity of the control). Macrosystems arising in social and economic sciences or biological sciences seem to exhibit heavy trajectories. We make explicit the differential equations providing slow and heavy trajectories when the viability domain is smooth

Qualitative Equations: The Confluence Case

This paper deals with a domain of Artificial Intelligence known under the name of "qualitative simulation" or "qualitative physics", to which special volumes of "Artificial Intelligence" (1984) and or "IEEE Transactions on Systems, Man and Cybernetics" (1987) have been devoted. It defines the concept of "qualitative frame" of a set, which allows to introduce strict, large and dual confluence frames of a finite dimensional vector-space. After providing a rigorous definition of standard, lower and upper qualitative solutions in terms of confluences introduced by De Kleer, it provides a duality criterion for the existence of a strict standard solution to both linear and non linear equations. It also furnishes a dual characterization of the existence or upper and lower qualitative solutions to a linear equation. These theorems are extended to the case or "inclusions", where single-valued maps are replaced by set-valued maps. This may be useful for dealing with qualitative properties of maps which are not precisely known, or which are defined by a set of properties, a requirement which is at the heart of qualitative simulation

Study of Solutions to Differential Inclusions by the "Pipe Method"

A pipe of a differential inclusion is a set-valued map associating with each time t a subset P(t) of states which contains a trajectory of the differential inclusion for any initial state x_o belonging to P(O). As in the Liapunov method, knowledge of a pipe provides information on the behavior of the trajectory. In this paper, the characterization of pipes and non-smooth analysis of set-valued maps are used to describe several classes of pipes. This research was conducted within the framework of the Dynamics of Macrosystems study in the System and Decision Sciences Program

Contingent Isaacs Equations of a Differential Game

The purpose of this paper is to characterize classical and lower semicontinuous solutions to the Hamilton-Jacobi-Isaacs partial differential equations associated with a differential game and, in particular, characterize closed subsets the indicators of which are solutions to these equations. For doing so, the classical concept of derivative is replaced by contingent epi-derivative, which can apply to any function. The use of indicator of subsets which are solutions of either one of the contingent Isaacs equation allows to characterize areas of the playability set in which some behavior (playability, winability, etc.) of the players can be achieved

Hazy Differential Inclusions

This paper is devoted to differential inclusions the right-hand sides of which are hazy subsets, which are fuzzy subsets whose membership functions are cost functions taking their values in [0,infinity] instead of [0,1]. By doing so, the concept of uncertainty involved in differential inclusions becomes more precise, by allowing the velocities not only to depend in a multivalued way upon the state of the system, but also in a fuzzy way. The viability theorems are adapted to hazy differential inclusions and to sets of state constraints which are either usual or hazy. The existence of a largest closed hazy viability domain contained in a given closed hazy subset is also provided

A Viability Approach to Liapunov's Second Method

The purpose of this note is to extend Liapunov's second method to the case of differential inclusions, when viability requirements are made and when the Liapunov functions are continuous