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

${\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)}$

where ${\displaystyle q_{0}(x)\neq 0}$ and ${\displaystyle q_{2}(x)\neq 0}$. If ${\displaystyle q_{0}(x)=0}$ the equation reduces to a Bernoulli equation, while if ${\displaystyle q_{2}(x)=0}$ 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

${\displaystyle y'=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}\!}$

then, wherever ${\displaystyle q_{2}}$ is non-zero and differentiable, ${\displaystyle v=yq_{2}}$ satisfies a Riccati equation of the form

${\displaystyle v'=v^{2}+R(x)v+S(x),\!}$

where ${\displaystyle S=q_{2}q_{0}}$ and ${\displaystyle R=q_{1}+{\frac {q_{2}'}{q_{2}}}}$, because

${\displaystyle v'=(yq_{2})'=y'q_{2}+yq_{2}'=(q_{0}+q_{1}y+q_{2}y^{2})q_{2}+v{\frac {q_{2}'}{q_{2}}}=q_{0}q_{2}+\left(q_{1}+{\frac {q_{2}'}{q_{2}}}\right)v+v^{2}.\!}$

Substituting ${\displaystyle v=-u'/u}$, it follows that ${\displaystyle u}$ satisfies the linear second-order ODE

${\displaystyle u''-R(x)u'+S(x)u=0\!}$

since

${\displaystyle v'=-(u'/u)'=-(u''/u)+(u'/u)^{2}=-(u''/u)+v^{2}\!}$

so that

${\displaystyle u''/u=v^{2}-v'=-S-Rv=-S+Ru'/u\!}$

and hence

${\displaystyle u''-Ru'+Su=0.\!}$

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

${\displaystyle y=-q_{2}^{-1}(\log(c_{1}u_{1}+c_{2}u_{2}))'.}$

Application to the Schwarzian equation

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation

${\displaystyle S(w):=(w''/w')'-(w''/w')^{2}/2=f}$

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 ${\displaystyle S(w)}$ has the remarkable property that it is invariant under Möbius transformations, i.e. ${\displaystyle S((aw+b)/(cw+d))=S(w)}$ whenever ${\displaystyle ad-bc}$ is non-zero.) The function ${\displaystyle y=w''/w'}$ satisfies the Riccati equation

${\displaystyle y'=y^{2}/2+f.}$

By the above ${\displaystyle y=-2u'/u}$ where ${\displaystyle u}$ is a solution of the linear ODE

${\displaystyle u''+(1/2)fu=0.}$

Since ${\displaystyle w''/w'=-2u'/u}$, integration gives ${\displaystyle w'=C/u^{2}}$ for some constant ${\displaystyle C}$. On the other hand any other independent solution ${\displaystyle U}$ of the linear ODE has constant non-zero Wronskian ${\displaystyle U'u-Uu'}$ which can be taken to be ${\displaystyle C}$ after scaling. Thus

${\displaystyle w'=(U'u-Uu')/u^{2}=(U/u)'}$

so that the Schwarzian equation has solution ${\displaystyle w=U/u.}$

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 ${\displaystyle y_{1}}$ can be found, the general solution is obtained as

${\displaystyle y=y_{1}+u}$

Substituting

${\displaystyle y_{1}+u}$

in the Riccati equation yields

${\displaystyle y_{1}'+u'=q_{0}+q_{1}\cdot (y_{1}+u)+q_{2}\cdot (y_{1}+u)^{2},}$

and since

${\displaystyle y_{1}'=q_{0}+q_{1}\,y_{1}+q_{2}\,y_{1}^{2},}$

it follows that

${\displaystyle u'=q_{1}\,u+2\,q_{2}\,y_{1}\,u+q_{2}\,u^{2}}$

or

${\displaystyle u'-(q_{1}+2\,q_{2}\,y_{1})\,u=q_{2}\,u^{2},}$

which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is

${\displaystyle z={\frac {1}{u}}}$

Substituting

${\displaystyle y=y_{1}+{\frac {1}{z}}}$

directly into the Riccati equation yields the linear equation

${\displaystyle z'+(q_{1}+2\,q_{2}\,y_{1})\,z=-q_{2}}$

A set of solutions to the Riccati equation is then given by

${\displaystyle y=y_{1}+{\frac {1}{z}}}$

where z is the general solution to the aforementioned linear equation.

See also

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

References

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.

Further reading

• Hille, Einar (1997) [1976], Ordinary Differential Equations in the Complex Domain, New York: Dover Publications, ISBN   0-486-69620-0
• 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
• Reid, William T. (1972), Riccati Differential Equations, London: Academic Press

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.