# Riccati Equation

Also found in: Wikipedia.

## Riccati equation

[ri′käd·ē i‚kwā·zhən]
(mathematics)
A first-order differential equation having the form y ′ = A0(x) + A1(x) y + A2(x) y 2; every second-order linear differential equation can be transformed into an equation of this form.
A matrix equation of the form dP (t)/ dt + P (t) F (t) + F T (t) P (t) -P (t) G (t) R -1(t) G T (t) P (t) + Q (t) = 0, which frequently arises in control and estimation theory.

## Riccati Equation

a first-order ordinary differential equation of the form

where a, b, and a are constants. This equation was first investigated by J. Riccati in 1724; certain special cases were studied earlier. In work done in 1724 and 1725, D. Bernoulli established that equation (*) can be integrated in terms of elementary functions when α = – 2 or α = – 4k/(2k – 1), where k is an integer. J. Liouville proved in 1841 that, for other values of α, the equation (*) cannot be solved by quadrature, that is, by applying a finite number of algebraic operations, transformations of variables, and indefinite integrations to elementary functions; a general solution can be expressed in terms of cylindrical functions.

The differential equation

where P(x), Q(x), and R(x) are continuous functions, is called the generalized Riccati equation. When P(x) = 0, the generalized Riccati equation is a linear differential equation; when R(x) = 0, it is the Bernoulli equation. In these two cases, the equation is integrable in closed form. Other cases of the integrability of the generalized Riccati equation have also been studied.

### REFERENCE

Kamke, E. Spravochnik po obyknovennym differentsial’nym uravneniiam, 4th ed. Moscow, 1971. (Translated from German.)
Mentioned in ?
References in periodicals archive ?
Therefore we can conclude here that there is neat linkage between Schrodinger equation, Maxwell equation, Riccati equation via biquaternion expression [22, 23, 24].
In order to regularize the LQ optimal control problem and guarantee the existence of a stabilizing solution to the associated Riccati equation, we modify the performance index as follows:
We seek a solution of this nonlinear Riccati equation in the form
The stability of the stochastic augmented error system is studied and the sufficient and necessary criteria which guarantee that there exists a unique semipositive definite solution to the corresponding Riccati equation have been met.
Our concern is a special kind of nonsymmetric algebraic Riccati equation (NARE) that irises in transport theory.
Algebraic Riccati equation of formula (11) can be translated as follows:
i] positive definite computed through Riccati equation.
The solution of these finite-dimensional problems can be reduced to the solution of a matrix Riccati equation.
MEHRMANN, A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation, IEEE Trans.
measure chains, Riccati equation, oscillation, nonoscillation, time scales
computed] of the Riccati equation degrades, until it finally stagnates by an error of the order of [10.
The linear-quadratic Gaussian problem of control theory, also known as the quadratic regulator problem, can be solved by solving an algebraic Riccati equation.

Site: Follow: Share:
Open / Close