Riccati equation

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In mathematics, a Riccati equation is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form

$y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x)$

where $q_0(x) \neq 0$ and $q_2(x) \neq 0$. If $q_0(x) = 0$ the equation reduces to a Bernoulli equation, while if $q_2(x) = 0$ 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.

Reduction to a second order linear equation

The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE) (Ince 1956, pp. 23–25): If

$y'=q_0(x) + q_1(x)y + q_2(x)y^2\!$

then, wherever $q_2$ is non-zero and differentiable, $v=yq_2$ satisfies a Riccati equation of the form

$v'=v^2 + R(x)v +S(x),\!$

where $S=q_2q_0$ and $R=q_1+\left(\frac{q_2'}{q_2}\right)$, because

$v'=(yq_2)'= y'q_2 +yq_2'=(q_0+q_1 y + q_2 y^2)q_2 + v \frac{q_2'}{q_2}=q_0q_2 +\left(q_1+\frac{q_2'}{q_2}\right) v + v^2.\!$

Substituting $v=-u'/u$, it follows that $u$ satisfies the linear 2nd order ODE

$u''-R(x)u' +S(x)u=0 \!$

since

$v'=-(u'/u)'=-(u''/u) +(u'/u)^2=-(u''/u)+v^2\!$

so that

$u''/u= v^2 -v'=-S -Rv=-S +Ru'/u\!$

and hence

$u'' -Ru' +Su=0.\!$

A solution of this equation will lead to a solution $y=-u'/(q_2u)$ of the original Riccati equation.

Application to the Schwarzian equation

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation

$S(w):=(w''/w')' - (w''/w')^2/2 =f$

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 $S(w)$ has the remarkable property that it is invariant under Möbius transformations, i.e. $S((aw+b)/(cw+d))=S(w)$ whenever $ad-bc$ is non-zero.) The function $y=w''/w'$ satisfies the Riccati equation

$y'=y^2/2 +f.$

By the above $y=-2u'/u$ where $u$ is a solution of the linear ODE

$u''+ (1/2) fu=0.$

Since $w''/w'=-2u'/u$, integration gives $w'=C /u^2$ for some constant $C$. On the other hand any other independent solution $U$ of the linear ODE has constant non-zero Wronskian $U'u-Uu'$ which can be taken to be $C$ after scaling. Thus

$w'=(U'u-Uu')/u^2=(U/u)'$

so that the Schwarzian equation has solution $w=U/u.$

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 $y_1$ can be found, the general solution is obtained as

$y = y_1 + u$

Substituting

$y_1 + u$

in the Riccati equation yields

$y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,$

and since

$y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2$
$u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2$

or

$u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2,$

which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is

$z =\frac{1}{u}$

Substituting

$y = y_1 + \frac{1}{z}$

directly into the Riccati equation yields the linear equation

$z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2$

A set of solutions to the Riccati equation is then given by

$y = y_1 + \frac{1}{z}$

where z is the general solution to the aforementioned linear equation.

References

1. ^ 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.

Further reading

• Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN 0-486-69620-0
• Ince, E. L. (1956) [1926], Ordinary Differential Equations, New York: Dover Publications
• Nehari, Zeev (1975) [1952], Conformal Mapping, New York: Dover Publications, ISBN 0-486-61137-X
• Polyanin, Andrei D.; Zaitsev, Valentin F. (2003), Handbook of Exact Solutions for Ordinary Differential Equations (2nd ed.), Boca Raton, Fla.: Chapman & Hall/CRC, ISBN 1-58488-297-2
• Zelikin, Mikhail I. (2000), Homogeneous Spaces and the Riccati Equation in the Calculus of Variations, Berlin: Springer-Verlag