Riccati equation

Riccati equations

Riccati equations


A (general) Riccati differential equation is any first-order ordinary differential equation for a function y=y(t)y = y(t) which is of the form

dydt=a 0(t)+a 1(t)y+a 2(t)y 2 \frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2

with a 0(t)0a_0(t)\neq 0 and a 2(t)0a_2(t)\neq 0.

Connection to fractional linear transformations

Riccati equations are to fractional linear transformations as linear first-order ordinary differential equations are to affine transformations, in the following sense. A linear first-order ordinary differential equation

dydt=a 0(t)+a 1(t)y \frac{d y}{d t} = a_0(t) + a_1(t) y

has (under suitable regularity conditions on a 0a_0 and a 1a_1) solutions y(t)y(t) depending in an affine manner on the initial data:

y(t)=f(t)y(0)+g(t) y(t) = f(t) y(0) + g(t)

for functions f,gf,g independent of y(0)y(0). More generally, any Riccati equation

dydt=a 0(t)+a 1(t)y+a 2(t)y 2 \frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2

has (under suitable regularity conditions, and at least for short times) solutions y(t)y(t) depending on the initial data as follows:

y(t)=f(t)y(0)+g(t)h(t)y(0)+i(t) y(t) = \frac{f(t) y(0) + g(t)}{h(t) y(0) + i(t)}

for functions f,g,h,if,g,h,i independent of y(0)y(0).

Unlike solutions of first-order linear equations, solutions of Riccati equations can blow up in finite time; however, these solutions become defined for all time if we let yy take values in the projective line rather than the affine line: that is, P 1\mathbb{R}\mathrm{P}^1 or P 1\mathbb{C}\mathrm{P}^1 rather than \mathbb{R} or \mathbb{C}.

Indeed, the group of fractional linear transformations, the projective general linear group PGL(2,)\mathrm{PGL}(2,\mathbb{R}) or PGL(2,)\mathrm{PGL}(2,\mathbb{C}), acts naturally on the projective line, with the group of affine transformations being the subgroup that preserves the affine line.

The group of smooth PGL(2)\mathrm{PGL}(2)-valued functions on the real line acts on the space of Riccati equations. First, note that this group acts on 1\mathbb{P}^1-valued functions on the real line as follows:

(f(t) g(t) h(t) i(t)):y(t)y˜(t)=f(t)y(t)+g(t)h(t)y(t)+i(t) \left( \begin{array}{cc} f(t) & g(t) \\ h(t) & i(t) \end{array} \right) \colon y(t) \mapsto \tilde{y}(t) = \frac{f(t) y(t) + g(t)}{h(t) y(t) + i(t)}

Then, given any Riccati equation

dydt=a 0(t)+a 1(t)y+a 2(t)y 2 \frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2 \qquad \star

there is another Riccati equation such that y˜\tilde{y} is a solution of this new Riccati equation iff yy is a solution of \star.

Correspondence with 2nd order linear ODEs

There is a correspondence between Riccati equations and a wide class of 2nd order linear differential equations, namely Riccati equation for yy gives a 2nd order linear ODE for uu which satisfies a 2u=uya_2 u' = u y, namely

u+a 0a 2u+(a 1+a 2a 2)u=0. u'' + a_0 a_2 u' + \left( a_1 + \frac{a_2'}{a_2} \right) u = 0.

Of course, the coefficients are non-constant.

Reduction to a Bernoulli ODE in the presence of a particular solution

Unlike the Bernoulli differential equation? which has a 0(z)=0a_0(z) = 0 and a 2(z)0a_2(z) \neq 0, the Riccati equation can not be solved in quadratures in general, unless a a particular solution y 1y_1 is known. If so, then we can write y=y 1+uy = y_1 + u and write down the equation for uu which appears to be an example of a Bernoulli differential equation, hence solvable in quadratures. Regarding that w=1/uw = 1/u is the substitution for reducing Bernoulli to first order linear ODE, we can reduce the original Riccati to such ODE by substituting y=y 1+1/wy = y_1 + 1/w.

Matrix Riccati equation

For a generalization, the matrix Riccati equation, see the eom article.


category: analysis

Last revised on September 7, 2020 at 12:54:36. See the history of this page for a list of all contributions to it.