Retro-Prospective Differential Inclusions and their Control by the Differential Connection Tensors of their Evolutions: The trendometer

TL;DR

This paper introduces a novel framework combining retrospective and prospective derivatives in differential inclusions, with applications to complex network systems and control via differential connection tensors.

Contribution

It proposes the concept of differential connection tensors to analyze and control evolutionary systems considering both retrospective and prospective derivatives.

Findings

Defined differential connection tensors for network junctions.

Applied the framework to control uncertain evolutionary systems.

Highlighted the importance of retrospective derivatives in non-smooth dynamics.

Abstract

This study is motivated by two different, yet, connected, motivations. The first one follows the observation that the classical definition of derivatives involves prospective (or forward) difference quotients, not known whenever the time is directed, at least at the macroscopic level. Actually, the available and known derivatives are retrospective (or backward). They co\"incide whenever the functions are differentiable in the classical sense, but not in the case of non smooth maps, single-valued or set-valued. The later ones are used in differential inclusions (and thus, in uncertain control systems) governing evolutions in function of time and state. We follow the plea of some physicists for taking also into account the retrospective derivatives to study prospective evolutions in function of time, state and retrospective derivatives, a particular, but specific, example of historical of ''path dependent'' evolutionary systems. This is even more crucial in life sciences, in the absence of experimentation of uncertain evolutionary systems. The second motivation emerged from the study of networks with junctions (cross-roads in traffic networks, synapses in neural networks, banks in financial networks, etc.), an important feature of ''complex systems''. At each junction, the velocities of the incoming (retrospective) and outgoing (prospective) evolutions are confronted. One measure of this confrontation (''jerkiness'') is provided by the product of the retrospective and prospective velocities, negative in ''inhibitory'' junctions, positive for ''excitatory'' ones, for instance. This leads to the introduction of the ''differential connection tensor'' of two evolutions, defined as the tensor product of retrospective and prospective derivatives, which can be used for controlling evolutionary systems governing the evolutions through networks with junctions.