Riccati Equation

Riccati equation [ri′käd·ē i‚kwā·zhən]

(mathematics)

A first-order differential equation having the formy′ =A₀(x) +A₁(x)y+A₂(x)y²; every second-order linear differential equation can be transformed into an equation of this form.

A matrix equation of the formdP(t)/dt+P(t)F(t) +F^T(t)P(t) -P(t)G(t)R^-1(t)G^T(t)P(t) +Q(t) = 0, which frequently arises in control and estimation theory.

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Riccati Equation 

a first-order ordinary differential equation of the form

where a, b, and a are constants. This equation was first investigated by J. Riccati in 1724; certain special cases were studied earlier. In work done in 1724 and 1725, D. Bernoulli established that equation (*) can be integrated in terms of elementary functions when α = – 2 or α = – 4k/(2k – 1), where k is an integer. J. Liouville proved in 1841 that, for other values of α, the equation (*) cannot be solved by quadrature, that is, by applying a finite number of algebraic operations, transformations of variables, and indefinite integrations to elementary functions; a general solution can be expressed in terms of cylindrical functions.

The differential equation

where P(x), Q(x), and R(x) are continuous functions, is called the generalized Riccati equation. When P(x) = 0, the generalized Riccati equation is a linear differential equation; when R(x) = 0, it is the Bernoulli equation. In these two cases, the equation is integrable in closed form. Other cases of the integrability of the generalized Riccati equation have also been studied.

REFERENCE

Kamke, E. Spravochnik po obyknovennym differentsial’nym uravneniiam, 4th ed. Moscow, 1971. (Translated from German.)