Riccati Equation

Also found in: Wikipedia.

Riccati equation

[ri′käd·ē i‚kwā·zhən]
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.


Kamke, E. Spravochnik po obyknovennym differentsial’nym uravneniiam, 4th ed. Moscow, 1971. (Translated from German.)
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.