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)

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):^[2] 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

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

↑ 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.
↑ Ince, E. L. (1956) [1926], Ordinary Differential Equations, New York: Dover Publications, pp. 23–25

External links

Hazewinkel, Michiel, ed. (2001), "Riccati equation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Riccati Equation at EqWorld: The World of Mathematical Equations.
Riccati Differential Equation at Mathworld
MATLAB function for solving continuous-time algebraic Riccati equation.
SciPy has functions for solving the continuous algebraic Riccati equation and the discrete algebraic Riccati equation.

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