On the controllability of distributed systems

To “control” a system is to make it behave (hopefully) according to our “wishes,” in a way compatible with safety and ethics, at the least possible cost. The systems considered here are distributed—i.e., governed (modeled) by partial differential equations (PDEs) of evolution. Our “wish” is to drive the system in a given time, by an adequate choice of the controls, from a given initial state to a final given state, which is the target. If this can be achieved (respectively, if we can reach any “neighborhood” of the target) the system, with the controls at our disposal, is exactly (respectively, approximately) controllable. A very general (and fuzzy) idea is that the more a system is “unstable” (chaotic, turbulent) the “simplest,” or the “cheapest,” it is to achieve exact or approximate controllability. When the PDEs are the Navier–Stokes equations, it leads to conjectures, which are presented and explained. Recent results, reported in this expository paper, essentially prove the conjectures in two space dimensions. In three space dimensions, a large number of new questions arise, some new results support (without proving) the conjectures, such as generic controllability and cases of decrease of cost of control when the instability increases. Short comments are made on models arising in climatology, thermoelasticity, non-Newtonian fluids, and molecular chemistry. The Introduction of the paper and the first part of all sections are not technical. Many open questions are mentioned in the text.

Gestational drive and the green-bearded placenta.

A "green beard" refers to a gene, or group of genes, that is able to recognize itself in other individuals and direct benefits to these individuals. Green-beard effects have been dismissed as implausible by authors who have implicitly assumed sophisticated mechanisms of perception and complex behavioral responses. However, many simple mechanisms for genes to "recognize" themselves exist at the maternal-fetal interface of viviparous organisms. Homophilic cell adhesion molecules, for example, are able to interact with copies of themselves on other cells. Thus, the necessary components of a green-beard effect -- feature, recognition, and response -- can be different aspects of the phenotype of a single gene. Other green-beard effects could involve coalitions of genes at closely linked loci. In fact, any form of epistasis between a locus expressed in a mother and a closely linked locus expressed in the fetus has the property of "self-recognition." Green-beard effects have many formal similarities to systems of meiotic drive and, like them, can be a source of intragenomic conflict.