Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Riccati equation
Summary
In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form
where and . If the equation reduces to a Bernoulli equation, while if the equation becomes a first order linear ordinary differential equation.
The equation is named after Jacopo Riccati (1676–1754).
More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.
The non-linear Riccati equation can always be converted to a second order linear ordinary differential equation (ODE):
If
then, wherever is non-zero and differentiable, satisfies a Riccati equation of the form
where and , because
Substituting , it follows that satisfies the linear 2nd order ODE
since
so that
and hence
A solution of this equation will lead to a solution of the original Riccati equation.
An important application of the Riccati equation is to the 3rd order Schwarzian differential equation
which occurs in the theory of conformal mapping and univalent functions. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative has the remarkable property that it is invariant under Möbius transformations, i.e. whenever is non-zero.) The function
satisfies the Riccati equation
By the above where is a solution of the linear ODE
Since , integration gives
for some constant . On the other hand any other independent solution of the linear ODE
has constant non-zero Wronskian which can be taken to be after scaling.
Thus
so that the Schwarzian equation has solution
The correspondence between Riccati equations and second-order linear ODEs has other consequences. For example, if one solution of a 2nd order ODE is known, then it is known that another solution can be obtained by quadrature, i.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.