Riccati equation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 Count Jacopo Francesco 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[edit]

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[edit]

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[edit]

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.

See also[edit]

External links[edit]

References[edit]

  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.
  • 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