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.

Conversion to a second order linear equation

The non-linear Riccati equation can always be converted 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 $S=q_{2}q_{0}$ and $R=q_{1}+{\tfrac {q_{2}'}{q_{2}}},$ because ${\begin{aligned}v'&=(yq_{2})'\\[4pt]&=y'q_{2}+yq_{2}'\\&=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+v{\frac {q_{2}'}{q_{2}}}\\&=q_{0}q_{2}+\left(q_{1}+{\frac {q_{2}'}{q_{2}}}\right)v+v^{2}\end{aligned}}$ Substituting $v=-{\tfrac {u'}{u}},$ it follows that $u$ satisfies the linear second-order ODE $u''-R(x)u'+S(x)u=0$ since ${\begin{aligned}v'&=-\left({\frac {u'}{u}}\right)'\\[2pt]&=-\left({\frac {u''}{u}}\right)+\left({\frac {u'}{u}}\right)^{2}\\[2pt]&=-\left({\frac {u''}{u}}\right)+v^{2}\end{aligned}}$ so that ${\begin{aligned}{\frac {u''}{u}}&=v^{2}-v'\\&=-S-Rv\\&=-S+R{\frac {u'}{u}}\end{aligned}}$ and hence $u''-Ru'+Su=0.$

Then substituting the two solutions of this linear second order equation into the transformation $y=-{\frac {u'}{q_{2}u}}=-q_{2}^{-1}{\bigl (}\log(u){\bigr )}'$ suffices to have global knowledge of the general solution of the Riccati equation by the formula:^[3] $y=-q_{2}^{-1}{\bigl (}\log(c_{1}u_{1}+c_{2}u_{2}){\bigr )}'.$

Application to the Schwarzian equation

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation $S(w):=\left({\frac {w''}{w'}}\right)'-{\frac {1}{2}}\left({\frac {w''}{w'}}\right)^{2}=f$ 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 $S (w)$ has the remarkable property that it is invariant under Möbius transformations, i.e. $S{\bigl (}{\tfrac {aw+b}{cw+d}}{\bigr )}=S(w)$ whenever $ad-bc$ is non-zero.) The function $y={\tfrac {w''}{w'}}$ satisfies the Riccati equation $y'={\frac {1}{2}}y^{2}+f.$ By the above $y=-2{\tfrac {u'}{u}}$ where $u$ is a solution of the linear ODE $u''+{\frac {1}{2}}fu=0.$ Since ${\tfrac {w''}{w'}}=-2{\tfrac {u'}{u}},$ integration gives $w'={\tfrac {C}{u^{2}}}$ 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 $w'={\frac {U'u-Uu'}{u^{2}}}=\left({\frac {U}{u}}\right)'$ so that the Schwarzian equation has solution $w={\tfrac {U}{u}}.$

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 $y_{1}'=q_{0}+q_{1}\,y_{1}+q_{2}\,y_{1}^{2},$ it follows that $u'=q_{1}\,u+2\,q_{2}\,y_{1}\,u+q_{2}\,u^{2}$ 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.

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
^ Conte, Robert (1999). "The Painlevé Approach to Nonlinear Ordinary Differential Equations". The Painlevé Property. New York, NY: Springer New York. pp. 5, 98. doi:10.1007/978-1-4612-1532-5_3. ISBN 978-0-387-98888-7.

External links

"Riccati equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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.

[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.

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

[3] Conte, Robert (1999). "The Painlevé Approach to Nonlinear Ordinary Differential Equations". The Painlevé Property. New York, NY: Springer New York. pp. 5, 98. doi:10.1007/978-1-4612-1532-5_3. ISBN 978-0-387-98888-7.

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