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

where and . If the equation reduces to a Bernoulli equation, while if the equation becomes a first order linear ordinary differential equation.

The equation is named after Jacopo Riccati (1676–1754)Riccati, Jacopo (1724) "Animadversiones in aequationes differentiales secundi gradus" (Observations regarding differential equations of the second order), Actorum Eruditorum, quae Lipsiae publicantur, Supplementa, 8 : 66-73. Translation of the original Latin into English by Ian Bruce. 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.

Conversion to a second order linear equation

The quadratic non-linear ordinary differential equation (ODE) known as the Riccati equation can be converted to a second order linear ordinary differential equation.

First, we show that there is a standard form where the Riccati equation's quadratic coefficient is unity. For a given , wherever is non-zero and differentiable, we may substitute , and find

with and .

Given this standard form, let us suppose there exists an unknown function such that . Then Thus, the solution of of a Riccati equation can be derived from the solution of the system

If the general solution of the linear equation is known, then

suffices to have global knowledge of the general solution of the Riccati equation by the formula:

Conversely, second order linear homogeneous ODEs link the Bessel equation can be transformed to a system equations based on a Riccati equation.

Complex analysis In complex analysis, the Riccati equation occurs as the first-order nonlinear ODE in the complex plane of the form

where and are polynomials in and locally analytic functions of , i.e., is a complex rational function. The only equation of this form that is of Painlevé type, is the Riccati equation

where are (possibly matrix) functions of .

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 has the remarkable property that it is invariant under Möbius transformations, i.e. whenever is non-zero.) The function satisfies the Riccati equation

By the above where is a solution of the linear ODE

Since integration gives for some constant . On the other hand any other independent solution of the linear ODE has constant non-zero Wronskian which can be taken to be after scaling. Thus

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 can be found, the general solution is obtained as

Substituting

in the Riccati equation yields

and since

it follows that

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 is the general solution to the aforementioned linear equation.

See also Linear-quadratic regulator Algebraic Riccati equation Linear-quadratic-Gaussian control

References

Riccati equation

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