Optimal Control of Fluid Forces Using Second Order Automatic Differentiation

Abstract

In resent years, the progress of computer and numerical computation technique allows not only complex numerical simulations but also resolution of the inverse analysis including optimal control problem. It is important to pursue high quality of the gradient computation in the inverse problem. Sometimes, it is called as the sensitivity analysis. In case of the inverse analysis, the minimization technique of a function is often used. Only first order variation of function is usually applied. But in order to minimize the functional exactly, it is also necessary to introduce the second order variation of the function. However, pursing variation is diffcult to formulate and to program. There is a method of computation to pursue the variation of functional automatically, i.e., Automatic Differentiation (AD). The AD is implemented with C++ programs using operator overloading technique. It computes the partial derivatives according to the differentiation rule of a composite function whenever basic operation is performed. Using AD, the first order variation can be obtained. Ordinary AD is called First Order Automatic Differentiation (FOAD). In addition, AD can be extended to the Second Order Automatic Differentiation (SOAD) to obtain second order variation. In this study, both FOAD and SOAD are applied to the optimal control problem of fluid forces. The control problem uses the minimization technique of the function. The purpose of this study is to present the application of SOAD in control problem. The Sakawa-Shindo Method using FOAD and the Steepest descent method using SOAD are compared as the minimization technique. The automatic differentiation is proposed as a new approach to the sensitivity analysis of the optimal control problem.