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

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 2002 Springer-Verlag New York
Subject
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
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-002-0749-z
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Dec 19, 2002

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