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 <math display=block> y'(x) = q_0(x) + q_1(x) \, y(x) + q_2(x) \, y^2(x) </math> where <math>q_0(x) \neq 0</math> and <math>q_2(x) \neq 0</math>. If <math>q_0(x) = 0</math> the equation reduces to a Bernoulli equation, while if <math>q_2(x) = 0</math> the equation becomes a first order linear ordinary differential equation.

The equation is named after Jacopo Riccati (1676–1754).<ref>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.</ref>

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):<ref>Template:Citation</ref> If <math display=block>y' = q_0(x) + q_1(x)y + q_2(x)y^2</math> then, wherever Template:Math is non-zero and differentiable, <math>v = yq_2</math> satisfies a Riccati equation of the form <math display=block>v' = v^2 + R(x)v + S(x),</math> where <math>S = q_2q_0</math> and <math>R = q_1 + \tfrac{q_2'}{q_2},</math> because <math display=block>\begin{align}

 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_0q_2  + \left(q_1+\frac{q_2'}{q_2}\right) v + v^2

\end{align}</math> Substituting <math>v = -\tfrac{u'}{u},</math> it follows that Template:Mvar satisfies the linear second-order ODE <math display=block>u - R(x)u' + S(x)u = 0</math> since <math display=block>\begin{align}

 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{align}</math> so that <math display=block>\begin{align}

 \frac{u}{u} &= v^2 - v' \\
 &= -S - Rv \\
 &= -S + R\frac{u'}{u}

\end{align}</math> and hence <math display=block>u - Ru' + Su = 0.</math>

Then substituting the two solutions of this linear second order equation into the transformation <math display=block>y = -\frac{u'}{q_2u} = -q_2^{-1} \bigl(\log(u) \bigr)'</math> suffices to have global knowledge of the general solution of the Riccati equation by the formula:<ref>Template:Cite book</ref> <math display=block>y = -q_2^{-1} \bigl(\log(c_1u_1 + c_2u_2) \bigr)'.</math>

Complex analysis

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

Application to the Schwarzian equation

An important application of the Riccati equation is to the 3rd order Schwarzian differential equation <math display=block>S(w) := \left(\frac{w}{w'}\right)' - \frac{1}{2}\left(\frac{w}{w'}\right)^2 = f</math> 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. <math>S\bigl(\tfrac{aw+b}{cw+d}\bigr) = S(w)</math> whenever <math>ad-bc</math> is non-zero.) The function <math>y = \tfrac{w}{w'}</math> satisfies the Riccati equation <math display=block>y' = \frac{1}{2}y^2 + f.</math> By the above <math>y = -2 \tfrac{u'}{u}</math> where Template:Mvar is a solution of the linear ODE <math display=block>u + \frac{1}{2}fu = 0.</math> Since <math> \tfrac{w}{w'} = -2\tfrac{u'}{u},</math> integration gives <math>w' = \tfrac{C}{u^2}</math> for some constant Template:Mvar. On the other hand any other independent solution Template:Mvar of the linear ODE has constant non-zero Wronskian <math>U'u - Uu'</math> which can be taken to be Template:Mvar after scaling. Thus <math display=block>w' = \frac{U'u-Uu'}{u^2} = \left(\frac{U}{u}\right)'</math> so that the Schwarzian equation has solution <math>w = \tfrac{U}{u}.</math>

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 <math display=block> y = y_1 + u </math> Substituting <math display=block> y_1 + u </math> in the Riccati equation yields <math display=block> y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,</math> and since <math display=block> y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2,</math> it follows that <math display=block> u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2 </math> or <math display=block> u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2, </math> which is a Bernoulli equation. The substitution that is needed to solve this Bernoulli equation is <math display=block> z =\frac{1}{u} </math> Substituting <math display=block> y = y_1 + \frac{1}{z} </math> directly into the Riccati equation yields the linear equation <math display=block> z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2 </math> A set of solutions to the Riccati equation is then given by <math display=block> y = y_1 + \frac{1}{z} </math> 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

• Template:Citation
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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.