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.

National policies for technical change: Where are the increasing returns to economic research?

Improvements over the past 30 years in statistical data, analysis, and related theory have strengthened the basis for science and technology policy by confirming the importance of technical change in national economic performance. But two important features of scientific and technological activities in the Organization for Economic Cooperation and Development countries are still not addressed adequately in mainstream economics: (i) the justification of public funding for basic research and (ii) persistent international differences in investment in research and development and related activities. In addition, one major gap is now emerging in our systems of empirical measurement—the development of software technology, especially in the service sector. There are therefore dangers of diminishing returns to the usefulness of economic research, which continues to rely completely on established theory and established statistical sources. Alternative propositions that deserve serious consideration are: (i) the economic usefulness of basic research is in the provision of (mainly tacit) skills rather than codified and applicable information; (ii) in developing and exploiting technological opportunities, institutional competencies are just as important as the incentive structures that they face; and (iii) software technology developed in traditional service sectors may now be a more important locus of technical change than software technology developed in “high-tech” manufacturing.