In
mathematics, a Riccati equation in the narrowest sense is any first-order
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
that is
quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''.
Mathematics ...
in the unknown function. In other words, it is an equation of the form
:
where
and
. If
the equation reduces to a
Bernoulli equation
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
, while if
the equation becomes a first order
linear ordinary differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
.
The equation is named after
Jacopo Riccati
Jacopo Francesco Riccati (28 May 1676 – 15 April 1754) was a Venetian mathematician and jurist from Venice. He is best known for having studied the equation which bears his name.
Education
Riccati was educated first at the Jesuit school for the ...
(1676–1754).
More generally, the term Riccati equation is used to refer to
matrix equations with an analogous quadratic term, which occur in both
continuous-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
and
discrete-time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
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
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = ...
(ODE):
If
:
then, wherever
is non-zero and differentiable,
satisfies a Riccati equation of the form
:
where
and
, because
:
Substituting
, it follows that
satisfies the linear 2nd order ODE
:
since
:
so that
:
and hence
:
A solution of this equation will lead to a solution
of the original Riccati equation.
Application to the Schwarzian equation
An important application of the Riccati equation is to the 3rd order
Schwarzian differential equation
:
which occurs in the theory of conformal mapping and
univalent functions
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.
Examples
The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f( ...
. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The
Schwarzian derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms a ...
has the remarkable property that it is invariant under Möbius transformations, i.e.
whenever
is non-zero.) The function
satisfies the Riccati equation
:
By the above
where
is a solution of the linear ODE
:
Since
, integration gives
for some constant
. On the other hand any other independent solution
of the linear ODE
has constant non-zero Wronskian
which can be taken to be
after scaling.
Thus
:
so that the Schwarzian equation has solution
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
can be found, the general solution is obtained as
:
Substituting
:
in the Riccati equation yields
:
and since
:
it follows that
:
or
:
which is a
Bernoulli equation
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
. The substitution that is needed to solve this Bernoulli equation is
:
Substituting
:
directly into the Riccati equation yields the linear equation
:
A set of solutions to the Riccati equation is then given by
:
where z is the general solution to the aforementioned linear equation.
See also
*
Linear-quadratic regulator
*
Algebraic Riccati equation
*
Linear-quadratic-Gaussian control
References
Further reading
*
*
*
*
*
External links
* {{springer, title=Riccati equation, id=p/r081770
Riccati Equationat EqWorld: The World of Mathematical Equations.
at
Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...
MATLAB functionfor solving continuous-time algebraic Riccati equation.
*
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal ...
has functions for solving th
continuous algebraic Riccati equationand th
Ordinary differential equations