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 $$y^{\prime }(x)=q_0(x)+q_1(x)y(x)+q_2(x)y^2(x)$$ where $q_0(x)\ne 0$ and $q_2(x)\ne 0$. 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) though John Bernoulli was the first to publish an equation of this form (1694) and James Bernoulli was the first to discover a solution (1703).

The term Riccati equation is now used very generally 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.

Conversion to a second order linear equation

The quadratic non-linear ordinary differential equation (ODE) known as the Riccati equation can be converted to a second order linear ordinary differential equation.

First, we show that there is a standard form where the Riccati equation's quadratic coefficient is unity. For a given $y^{\prime }=q_0(x)+q_1(x)y+q_2(x)y^2$, wherever $q_2$ is non-zero and differentiable, we may substitute $v=y{q}_2$, and find

$$\begin{array}{cc}{v}^{\prime }& =(y{q}_2{)}^{\prime }\\ & =y^{\prime }{q}_2+y{q_2}^{\prime }\\ & =(q_0+q_{1}y+q_2{y}^2)q_2+v\frac{{q_2}^{\prime }}{q_2}\\ & =q_0{q}_2+(q_1+\frac{{q_2}^{\prime }}{q_2})v+v^2\\ & =S(x)+R(x)v+v^2\end{array}$$ with $S=q_0{q}_2$ and $R=q_1+{q_2}^{\prime }/q_2$.

Given this standard form, let us suppose there exists an unknown function $u(x)$ such that $u^{\prime }=-uv$. Then $$\begin{array}{cc}{u}^{″}& =-u^{\prime }v-v^{\prime }u\\ & =v^{2}u-v^{\prime }u\\ & =v^{2}u-(S(x)+R(x)v+v^2)u\\ & =-uS(x)-R(x)uv\\ & =-uS(x)+R(x)u^{\prime }.\end{array}$$ Thus, the solution of $y(x)$ of a Riccati equation can be derived from the solution of the system $$\begin{array}{cc}y& =v/q_2\\ 0& =u^{\prime }+uv\\ 0& =u^{″}-R(x)u^{\prime }+S(x)u.\end{array}$$ If the general solution $u(x)$ of the linear equation is known, then $y=-\frac{u^{\prime }}{q_{2}u}=-q_2^{-1}{[\log (u)]}^{\prime }$ suffices to have global knowledge of the general solution of the Riccati equation by the formula: $$y=-q_2^{-1}{[\log (c_1{u}_1+c_2{u}_2)]}^{\prime }.$$

Conversely, second order linear homogeneous ODEs link the Bessel equation can be transformed to a system equations based on a Riccati equation.

Complex analysis

In complex analysis, the Riccati equation occurs as the first-order nonlinear ODE in the complex plane of the form $$\frac{dw}{dz}=F(w,z)=\frac{P(w,z)}{Q(w,z)},$$ where $P$ and $Q$ are polynomials in $w$ and locally analytic functions of $z\in ℂ$, i.e., $F$ is a complex rational function. The only equation of this form that is of Painlevé type, is the Riccati equation $$\frac{dw(z)}{dz}=A_0(z)+A_1(z)w+A_2(z)w^2,$$ where $A_i(z)$ are (possibly matrix) functions of $z$.

Application to the Schwarzian equation

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation $$S(w):={\left(\frac{w^{″}}{w^{\prime }}\right)}^{\prime }-\frac{1}{2}{\left(\frac{w^{″}}{w^{\prime }}\right)}^2=f$$ 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 Template:Math has the remarkable property that it is invariant under Möbius transformations, i.e. $S\left(\frac{aw+b}{cw+d}\right)=S(w)$ whenever $ad-bc$ is non-zero.) The function $y=\frac{w^{″}}{w^{\prime }}$ satisfies the Riccati equation $$y^{\prime }=\frac{1}{2}{y}^2+f.$$ By the above $y=-2\frac{u^{\prime }}{u}$ where Template:Mvar is a solution of the linear ODE $$u^{″}+\frac{1}{2}fu=0.$$ Since $\frac{w^{″}}{w^{\prime }}=-2\frac{u^{\prime }}{u},$ integration gives $w^{\prime }=\frac{C}{u^2}$ for some constant Template:Mvar. On the other hand any other independent solution Template:Mvar of the linear ODE has constant non-zero Wronskian $U^{\prime }u-U{u}^{\prime }$ which can be taken to be Template:Mvar after scaling. Thus $$w^{\prime }=\frac{U^{\prime }u-U{u}^{\prime }}{u^2}={\left(\frac{U}{u}\right)}^{\prime }$$ so that the Schwarzian equation has solution $w=\frac{U}{u}.$

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 Template:Math can be found, the general solution is obtained as $$y=y_1+u$$ Substituting $$y_1+u$$ in the Riccati equation yields $${y_1}^{\prime }+u^{\prime }=q_0+q_1\cdot (y_1+u)+q_2\cdot (y_1+u{)}^2,$$ and since $${y_1}^{\prime }=q_0+q_1{y}_1+q_2{y}_1^2,$$ it follows that $$u^{\prime }=q_{1}u+2q_2{y}_{1}u+q_2{u}^2$$ or $$u^{\prime }-(q_1+2q_2{y}_1)u=q_2{u}^2,$$ 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 $$z^{\prime }+(q_1+2q_2{y}_1)z=-q_2$$ A set of solutions to the Riccati equation is then given by $$y=y_1+\frac{1}{z}$$ where Template:Mvar is the general solution to the aforementioned linear equation.

See also

• Linear-quadratic regulator
• Algebraic Riccati equation
• Linear-quadratic-Gaussian control

Further reading

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External links

• Template:Springer
• 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.