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
Contents
- Reduction to a second order linear equation
- Application to the Schwarzian equation
- Obtaining solutions by quadrature
- References
where
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.
Reduction to a second order linear equation
The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE): If
then, wherever
where
Substituting
since
so that
and hence
A solution of this equation will lead to a solution
Application to the Schwarzian 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
By the above
Since
so that the Schwarzian equation has solution
Obtaining solutions by quadrature
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.e., a simple integration. The same holds true for the Riccati equation. In fact, if one particular solution
Substituting
in the Riccati equation yields
and since
or
which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is
Substituting
directly into the Riccati equation yields the linear equation
A set of solutions to the Riccati equation is then given by
where z is the general solution to the aforementioned linear equation.