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http://hdl.handle.net/10773/4109

title:  Twodimensional Newton's problem of minimal resistance 
authors:  Silva, C.J. Torres, D.F.M. 
keywords:  Calculus of variations Dimension two Newton's problem of minimal resistance Optimal control 
issue date:  2006 
publisher:  Polish Academy of Sciences 
abstract:  Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axissymmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well documented in the literature. Of course, at a first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and asked at least as many experts on the problem, and we have not been able to identify a single source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the twodimensional problem is richer than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classical threedimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem does not admit a local minimum), we prove that in dimension two the unrestricted problem is also wellposed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique solution, we show that in the restricted twodimensional problem the minimizer is not always unique  when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions. 
URI:  http://hdl.handle.net/10773/4109 
ISSN:  03248569 
publisher version/DOI:  http://control.ibspan.waw.pl:3000/contents/export?filename=SilvaTorres.pdf 
source:  Control and Cybernetics 
appears in collections  MAT  Artigos

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