Riccati equation
Encyclopedia
In mathematics
, a Riccati equation is any ordinary differential equation
that is quadratic
in the unknown function. In other words, it is an equation of the form
where
and 
(
is a Bernoulli equation
and
is a first order linear ordinary differential equation). It is named after Count Jacopo Francesco Riccati (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 and discrete-time linear-quadratic-Gaussian control
. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation
.
then, wherever
is non-zero, 
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.
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 this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative
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
, 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
or
which is a Bernoulli equation
. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Riccati equation is any ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
that is quadratic
Quadratic function
A quadratic function, in mathematics, is a polynomial function of the formf=ax^2+bx+c,\quad a \ne 0.The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis....
in the unknown function. In other words, it is an equation of the form
where
and ( is a Bernoulli equationBernoulli differential equation
In mathematics, an ordinary differential equation of the formis called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695...
and
is a first order linear ordinary differential equation). It is named after Count Jacopo Francesco Riccati (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 and discrete-time linear-quadratic-Gaussian control
Linear-quadratic-Gaussian control
In control theory, the linear-quadratic-Gaussian control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information and undergoing control subject to quadratic costs...
. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation
Algebraic Riccati equation
The algebraic Riccati equation is either of the following matrix equations:the continuous time algebraic Riccati equation :or the discrete time algebraic Riccati equation :...
.
Reduction to a second order linear equation
The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE) . Ifthen, wherever
is non-zero, satisfies a Riccati equation of the formwhere
and , becauseSubstituting
, it follows that satisfies the linear 2nd order ODEsince
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 this case the ODEs are in the complex domain and differentiation is with respect to a complex variable. (The Schwarzian derivative
Schwarzian derivative
In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and...
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 ODESince
, integration gives for some constant
. On the other hand any other independent solution of the linear ODEhas 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 quadratureNumerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
, 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 asSubstituting
in the Riccati equation yields
and since
or
which is a Bernoulli equation
Bernoulli differential equation
In mathematics, an ordinary differential equation of the formis called a Bernoulli equation when n≠1, 0, which is named after Jakob Bernoulli, who discussed it in 1695...
. 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 regulatorLinear-quadratic regulatorThe theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic functional is called the LQ problem...
- Algebraic Riccati equationAlgebraic Riccati equationThe algebraic Riccati equation is either of the following matrix equations:the continuous time algebraic Riccati equation :or the discrete time algebraic Riccati equation :...
- Matrix Riccati equation#Mathematical description of the problem and solution
External links
- Riccati Equation at EqWorld: The World of Mathematical Equations.
- Riccati Differential Equation at MathworldMathWorldMathWorld 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 Digital Library grant to the University of Illinois at...