Riccati

The Riccati equation is a type of first-order nonlinear ordinary differential equation named after the Italian mathematician Jacopo Riccati (1676–1754). In its standard form it is dy/dx = q0(x) + q1(x) y + q2(x) y^2, where q0, q1, q2 are given functions of x. The equation is simple in appearance but serves as a central example of nonlinear ODEs and arises in many areas of mathematics and applied science.

Solving the general Riccati equation typically involves transforming it into a linear problem. If q2(x) is not zero, the substitution y = -(u'/(q2(x) u)) converts it into a second-order linear differential equation for u. Alternatively, if a particular solution y_p is known, the substitution y = y_p + 1/v reduces the problem to a first-order linear equation for v.

Special cases include the linear case when q2 ≡ 0, which reduces to a first-order linear ODE. For specific coefficient choices, closed-form solutions can be obtained, and many numerical methods are used for the general case.

In control theory and filtering, the term Riccati equation is also used for the algebraic and differential Riccati equations. The differential form often appears as P' = A^T P + P A - P B R^{-1} B^T P + Q, with time dependence when A, B, Q, R vary in time; in the algebraic case, the same relation is solved for a steady matrix P. These equations are central in optimal control, estimation, and related areas.

Riccati-type equations also appear in physics, for example in the factorization method of quantum mechanics and in reductions of the Schrödinger equation, where they connect to supersymmetric formulations and to transformations of potentials.