Riccati equation

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

${\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)}$

where ${\displaystyle q_{0}(x)\neq 0}$ and ${\displaystyle q_{2}(x)\neq 0}$. If ${\displaystyle q_{0}(x)=0}$ the equation reduces to a Bernoulli equation, while if ${\displaystyle q_{2}(x)=0}$ the equation becomes a first order linear ordinary differential equation.

The equation is named after Jacopo Riccati (1676–1754).^[1]

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.