Averaging of Differential Equations Generating Oscillations and an Application to Control

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.