# Adjoint different equations of a system

1. Apr 15, 2013

### roldy

I'm working on an atmospheric reentry optimization problem that I found in a paper. The purpose is to find the control history for alpha (attack angle) that results in reaching the target range ($\phi$) within operational constraints and terminal constraints.

I'm using 3-DOF equations for velocity, range, and flight-path angle(θ) as well as 2 additional equations for a pilot penalty function (g-loading) and altitude penalty function (if capsule skips out above a predefined threshold, which it doesn't for this simulation). The pdf document that I am referencing is found here.

The problem I am encountering solving for the adjoint different equations (see page 30-33). Could someone explain what these are and how to get them? The paper says that I need to get information on how a change in the range, pilot penalty, and altitude penalty are related to $\delta \alpha$(t), the change in the control program and this is done by finding special solutions to the adjoint differential equations of the system. The paper has these differential adjoint equations but they don't mention how to find the initial value of the adjoint differential equation. The control program is updated as follows:

$\alpha_1(t) = \alpha_0(t) + \delta \alpha(t)$