Riccati equation
E649479
difference equation type
mathematical concept
nonlinear differential equation
ordinary differential equation type
A Riccati equation is a type of nonlinear differential or difference equation, often quadratic in the unknown function, that plays a central role in control theory, filtering, and various areas of applied mathematics.
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
difference equation type
ⓘ
mathematical concept ⓘ nonlinear differential equation ⓘ ordinary differential equation type ⓘ |
| appearsIn |
continuous-time optimal control problems
ⓘ
discrete-time optimal control problems ⓘ state estimation problems ⓘ |
| canBeTransformedTo | second-order linear differential equation ⓘ |
| field |
applied mathematics
ⓘ
control theory ⓘ differential equations ⓘ estimation theory ⓘ filtering theory ⓘ optimal control ⓘ stochastic control ⓘ |
| generalizes | Bernoulli differential equation NERFINISHED ⓘ |
| hasForm |
dy/dx = q0(x) + q1(x) y + q2(x) y^2
ⓘ
x_{k+1} = A_k + B_k x_k + C_k x_k^2 (difference form) ⓘ y'(x) = a(x) y(x)^2 + b(x) y(x) + c(x) ⓘ |
| hasVariant |
algebraic Riccati equation
ⓘ
differential Riccati equation NERFINISHED ⓘ discrete-time Riccati equation NERFINISHED ⓘ stochastic Riccati equation ⓘ |
| isNonlinearIn | unknown function ⓘ |
| isQuadraticIn | unknown function ⓘ |
| namedAfter | Jacopo Francesco Riccati NERFINISHED ⓘ |
| property |
first-order (differential form)
ⓘ
nonlinear ⓘ quadratic nonlinearity ⓘ |
| relatedTo |
Schrödinger equation
NERFINISHED
ⓘ
logarithmic derivative transformation ⓘ |
| solutionDependsOn | coefficients a(x), b(x), c(x) ⓘ |
| solutionSpace | can exhibit finite-time blow-up ⓘ |
| solvedBy |
linearization techniques
ⓘ
particular solution substitution ⓘ reduction to second-order linear ODE ⓘ |
| specialCaseOf | first-order nonlinear ordinary differential equation ⓘ |
| usedIn |
H-infinity control
ⓘ
Kalman filtering NERFINISHED ⓘ dynamic programming ⓘ filter design ⓘ financial mathematics ⓘ linear quadratic Gaussian control ⓘ linear quadratic regulator NERFINISHED ⓘ population dynamics models ⓘ quantum mechanics NERFINISHED ⓘ signal processing ⓘ smoothing algorithms ⓘ soliton theory ⓘ trajectory optimization ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Kailath factorization