Singularities and continuous dependence of the steady-state solution

In our research on aircraft guidance the desired control for automatic flight is computed in two parts. Stable inversion, which is used to find the correct input once the output trajectory is determined, is combined with a feedback controller. Stable inversion produces a control that is bounded for all time in response to an output that is similarly bounded. In many interesting cases, stable inversion evolves into finding a bounded and continuous solution of an ordinary differential equation (or system of differential equations) in response to a bounded excitation. Under the correct assumptions a unique solution exists that is bounded and continuous for all time. We indicate how this solution can be thought of as a "generalized" steady-state solution. Since this differential equation often depends on parameters that can change the order of the system, we want to study continuous dependence of the steady-state solution under such parameter changes. Our work is for nonlinear differential equations since our mathematical models of the aircraft are highly nonlinear, but the special case of linear equations is studied first and then applied to the nonlinear equations through an iterative process. For linear differential equations we study continuous dependence of Fourier coefficients, Fourier series, and then the steady state solution, which can be found by using forward and inverse Fourier transforms.