Averaging of Differential Equations Generating Oscillations and an Application to Control

Averaging of Differential Equations Generating Oscillations and an Application to Control Abstract. In this article we consider differential equations which generate oscillating solutions. These oscillations are due to the presence of a small parameter ɛ>0 ; however, they are not present in the coefficients but instead they are caused by a penalty term involving an antisymmetric operator. Our aims are twofold. In the first part we study asymptotics at all orders, for ɛ→ 0 , construct approximate solutions, and derive estimates of the error between the exact solution and the approximate ones. One of the motivations of this part is the study to high orders of the geostrophic asymptotics in atmospheric science, but there are many other possible applications involving in particular the wave equation. The actual applications of our results to atmospheric science will be discussed elsewhere [STW], as well as, on the mathematical side, the application to partial differential equations [TW1]. In the second part of this article we study a control problem involving such an equation and study the behavior of the state equation, of the optimal control, and of the optimality equation as ɛ→ 0 . For the control part we restrict ourselves to a linear equation and to the first order in the asymptotics ɛ→ 0 , leaving nonlinear problems and higher orders to a future work. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Averaging of Differential Equations Generating Oscillations and an Application to Control

Loading next page...
Copyright © Inc. by 2002 Springer-Verlag New York
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
Publisher site
See Article on Publisher Site

There are no references for this article.

You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial