NDLI: On the controllability of distributed systems

Content Provider	PubMed Central
Author	Lions, J. L.
Copyright Year	1997
Abstract	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.
Starting Page	4828
File Format	PDF
ISSN	10916490
e-ISSN	10916490
Journal	Proceedings of the National Academy of Sciences of the United States of America
Issue Number	10
Volume Number	94
Language	English
Publisher	The National Academy of Sciences of the USA
Publisher Date	1997-05-13
Access Restriction	Open
Rights Holder	The National Academy of Sciences of the USA
Subject Keyword	Research in Higher Education
Content Type	Text
Resource Type	Article
Subject	Multidisciplinary

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