From: prussing@aae.uiuc.edu (John Prussing) Subject: Re: continuity test Date: Thu, 16 Mar 2000 01:33:46 GMT Newsgroups: sci.math Summary: [missing] In <38CFF3EE.13EC0D86@brunel.ac.uk> "Johannes H. Andersen" writes: >John Prussing wrote: >> >> >> This type of continuity question arises in spacecraft trajectory problems, >> which some of your colleagues in aerospace engineering at PSU study. >> For an "impulsive" (delta "function") rocket thrust, the velocity >> vector is discontinuous. Some constants of the motion for an optimal >> (minimum-fuel) trajectory are functions of the velocity vector, so >> the question of continuity of these constants is important. >So the velocity jumps from v1 to v2 due to a short rocket burst. >In general it is unlikely that a function g(f(t)) of a discontinous >function f(t) is continous in t, unless g(u) is a constant funtion. >The discontinuity arises from the problem formulation, few real life >funtions are discontinous. Your optimal trajectory problem could >be an "optimal control" problem. Look at Pontryagin's book on >optimal control theory. >Johannes An optimal control problem is exactly what it is. The constants of the motion I mentioned come from the variational problem of minimizing a functional that represents the propellant consumed. And I oversimplified the problem in my question, but thanks for your comments. And yes, the velocity is not really discontinuous but, like a baseball being hit or a soccer ball being kicked, the change in velocity happens over such a small time interval (compared with the rest of the motion) that it can be idealized as zero time. -- =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= John E. Prussing Dept. of Aeronautical & Astronautical Engineering University of Illinois at Urbana-Champaign http://www.uiuc.edu/~prussing =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= ============================================================================== From: "Johannes H. Andersen" Subject: Re: continuity test Date: Thu, 16 Mar 2000 17:43:34 -0800 Newsgroups: sci.math John Prussing wrote: [quote of previous article deleted --djr] Control Theory is now a vast subject, there is classical, modern and even post-modern varieties. As a non-expert myself, I propose the book by Pontryagin, Bol'tanskii, Gamkreliedze, Mischenko: The Mathematical Theory of Optimal Processes. Pergamon 1964 as a starting point. The interesting contributions are precisely in discontinous, but piecewise continous control functions or forcing functions. Johannes ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: continuity test Date: 15 Mar 2000 15:47:55 -0600 Newsgroups: sci.math In article prussing@aae.uiuc.edu (John Prussing) writes: > This type of continuity question arises in spacecraft trajectory problems, > which some of your colleagues in aerospace engineering at PSU study. > For an "impulsive" (delta "function") rocket thrust, the velocity > vector is discontinuous. Some constants of the motion for an optimal > (minimum-fuel) trajectory are functions of the velocity vector, so > the question of continuity of these constants is important. Someone else mentioned that optimal control might be the place to look; he mentioned Pontrjagin's book, but I would suggest that more modern sources would possibly be even better. Some recent work has been done on addressing some issues related to discontinuities. I don't know if this work is relevant to your question, though. But if it is, SIAM and IEEE journals, and conference proceedings, might be the place to look. Kevin.