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Type of differential 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 ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)} where q 0 ( x ) ≠ 0 {\displaystyle q_{0}(x)\neq 0} and q 2 ( x ) ≠ 0 {\displaystyle q_{2}(x)\neq 0} . If q 0 ( x ) = 0 {\displaystyle q_{0}(x)=0} the equation reduces to a Bernoulli equation, while if q 2 ( x ) = 0 {\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[edit]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 {\displaystyle y'=q_{0}(x)+q_{1}(x)y+q_{2}(x)y^{2}} then, wherever q2 is non-zero and differentiable, v = y q 2 {\displaystyle v=yq_{2}} satisfies a Riccati equation of the form v ′ = v 2 + R ( x ) v + S ( x ) , {\displaystyle v'=v^{2}+R(x)v+S(x),} where S = q 2 q 0 {\displaystyle S=q_{2}q_{0}} and R = q 1 + q 2 ′ q 2 , {\displaystyle R=q_{1}+{\tfrac {q_{2}'}{q_{2}}},} because v ′ = ( y q 2 ) ′ = y ′ q 2 + y q 2 ′ = ( q 0 + q 1 y + q 2 y 2 ) q 2 + v q 2 ′ q 2 = q 0 q 2 + ( q 1 + q 2 ′ q 2 ) v + v 2 {\displaystyle {\begin{aligned}v'&=(yq_{2})'\\[4pt]&=y'q_{2}+yq_{2}'\\&=\left(q_{0}+q_{1}y+q_{2}y^{2}\right)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 = − u ′ u , {\displaystyle v=-{\tfrac {u'}{u}},} it follows that u satisfies the linear second-order ODE u ″ − R ( x ) u ′ + S ( x ) u = 0 {\displaystyle u''-R(x)u'+S(x)u=0} since v ′ = − ( u ′ u ) ′ = − ( u ″ u ) + ( u ′ u ) 2 = − ( u ″ u ) + v 2 {\displaystyle {\begin{aligned}v'&=-\left({\frac {u'}{u}}\right)'=-\left({\frac {u''}{u}}\right)+\left({\frac {u'}{u}}\right)^{2}\\[2pt]&=-\left({\frac {u''}{u}}\right)+v^{2}\end{aligned}}} so that u ″ u = v 2 − v ′ = − S − R v = − S + R u ′ u {\displaystyle {\begin{aligned}{\frac {u''}{u}}&=v^{2}-v'\\&=-S-Rv\\&=-S+R{\frac {u'}{u}}\end{aligned}}} and hence u ″ − R u ′ + S u = 0. {\displaystyle u''-Ru'+Su=0.}
Then substituting the two solutions of this linear second order equation into the transformation y = − u ′ q 2 u = − q 2 − 1 [ log ( u ) ] ′ {\displaystyle y=-{\frac {u'}{q_{2}u}}=-q_{2}^{-1}\left[\log(u)\right]'} suffices to have global knowledge of the general solution of the Riccati equation by the formula:[3] y = − q 2 − 1 [ log ( c 1 u 1 + c 2 u 2 ) ] ′ . {\displaystyle y=-q_{2}^{-1}\left[\log(c_{1}u_{1}+c_{2}u_{2})\right]'.}
In complex analysis, the Riccati equation occurs as the first-order nonlinear ODE in the complex plane of the form[4] d w d z = F ( w , z ) = P ( w , z ) Q ( w , z ) , {\displaystyle {\frac {dw}{dz}}=F(w,z)={\frac {P(w,z)}{Q(w,z)}},} where P {\displaystyle P} and Q {\displaystyle Q} are polynomials in w {\displaystyle w} and locally analytic functions of z ∈ C {\displaystyle z\in \mathbb {C} } , i.e., F {\displaystyle F} is a complex rational function. The only equation of this form that is of Painlevé type, is the Riccati equation d w ( z ) d z = A 0 ( z ) + A 1 ( z ) w + A 2 ( z ) w 2 , {\displaystyle {\frac {dw(z)}{dz}}=A_{0}(z)+A_{1}(z)w+A_{2}(z)w^{2},} where A i ( z ) {\displaystyle A_{i}(z)} are (possibly matrix) functions of z {\displaystyle z} .
Application to the Schwarzian equation[edit]An important application of the Riccati equation is to the 3rd order Schwarzian differential equation S ( w ) := ( w ″ w ′ ) ′ − 1 2 ( w ″ w ′ ) 2 = f {\displaystyle 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 ( a w + b c w + d ) = S ( w ) {\displaystyle S{\bigl (}{\tfrac {aw+b}{cw+d}}{\bigr )}=S(w)} whenever a d − b c {\displaystyle ad-bc} is non-zero.) The function y = w ″ w ′ {\displaystyle y={\tfrac {w''}{w'}}} satisfies the Riccati equation y ′ = 1 2 y 2 + f . {\displaystyle y'={\frac {1}{2}}y^{2}+f.} By the above y = − 2 u ′ u {\displaystyle y=-2{\tfrac {u'}{u}}} where u is a solution of the linear ODE u ″ + 1 2 f u = 0. {\displaystyle u''+{\frac {1}{2}}fu=0.} Since w ″ w ′ = − 2 u ′ u , {\displaystyle {\tfrac {w''}{w'}}=-2{\tfrac {u'}{u}},} integration gives w ′ = C u 2 {\displaystyle 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 − U u ′ {\displaystyle U'u-Uu'} which can be taken to be C after scaling. Thus w ′ = U ′ u − U u ′ u 2 = ( U u ) ′ {\displaystyle w'={\frac {U'u-Uu'}{u^{2}}}=\left({\frac {U}{u}}\right)'} so that the Schwarzian equation has solution w = U u . {\displaystyle w={\tfrac {U}{u}}.}
Obtaining solutions by quadrature[edit]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 y1 can be found, the general solution is obtained as y = y 1 + u {\displaystyle y=y_{1}+u} Substituting y 1 + u {\displaystyle y_{1}+u} in the Riccati equation yields y 1 ′ + u ′ = q 0 + q 1 ⋅ ( y 1 + u ) + q 2 ⋅ ( y 1 + u ) 2 , {\displaystyle 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 , {\displaystyle 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 {\displaystyle 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 , {\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 z = 1 u {\displaystyle z={\frac {1}{u}}} Substituting y = y 1 + 1 z {\displaystyle 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 {\displaystyle 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 + 1 z {\displaystyle y=y_{1}+{\frac {1}{z}}} where z is the general solution to the aforementioned linear equation.
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