On the controllability of distributed systems

J. L. Lions
1997 Proceedings of the National Academy of Sciences of the United States of America  
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
more » ... 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 f luids, 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.
doi:10.1073/pnas.94.10.4828 pmid:11038539 pmcid:PMC24591 fatcat:dct7caedv5htnbsbc5rzek2ytu