nLab Riccati equation

Idea

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

$$\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2$$

with $a_0(t)\neq 0$ and $a_2(t)\neq 0$.

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

$$\frac{d y}{d t} = a_0(t) + a_1(t) y$$

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

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

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

$$\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2$$

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

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

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

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 $y$ take values in the projective line rather than the affine line: that is, $\mathbb{R}\mathrm{P}^1$ or $\mathbb{C}\mathrm{P}^1$ rather than $\mathbb{R}$ or $\mathbb{C}$.

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

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

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

Then, given any Riccati equation

$$\frac{d y}{d t} = a_0(t) + a_1(t) y + a_2(t) y^2 \qquad \star$$

there is another Riccati equation such that $\tilde{y}$ is a solution of this new Riccati equation iff $y$ 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 $y$ gives a 2nd order linear ODE for $u$ which satisfies $a_2 u' = u y$, namely

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

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

Matrix Riccati equation

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

Literature

• Springer eom Riccati equation

• Wikipedia: Riccati equation

category: analysis