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 y ′ ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)} where q 0 ( x ) ≠ 0 {\displaystyle q_{0}(x)\neq 0} and q 2 ( x ) ≠ 0 {\displaystyle q_{2}(x)\neq 0} . If q 0 ( x ) = 0 {\displaystyle q_{0}(x)=0} the equation reduces to a Bernoulli equation, while if q 2 ( x ) = 0 {\displaystyle q_{2}(x)=0} the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754) though John Bernoulli was the first to publish an equation of this form (1694) and James Bernoulli was the first to discover a solution (1703). The term Riccati equation is now used very generally 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.