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 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 q 0 ( x ) ≠ 0 {displaystyle q_{0}(x) eq 0} and q 2 ( x ) ≠ 0 {displaystyle q_{2}(x) eq 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). 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 reduced to a second order linear ordinary differential equation (ODE):If then, wherever q 2 {displaystyle q_{2}} is non-zero and differentiable, v = y q 2 {displaystyle v=yq_{2}} satisfies a Riccati equation of the form where S = q 2 q 0 {displaystyle S=q_{2}q_{0}} and R = q 1 + ( q 2 ′ q 2 ) {displaystyle R=q_{1}+left({frac {q_{2}'}{q_{2}}} ight)} , because Substituting v = − u ′ / u {displaystyle v=-u'/u} , it follows that u {displaystyle u} satisfies the linear 2nd order ODE