Riccati equation

A first-order ordinary differential equation of the form

$$ \tag{1 } z ^ \prime + az ^ {2} = bt ^ \alpha , $$

where $ a $, $ b $ and $ \alpha $ are constants. This equation was first studied by J. Riccati (1723, see [1]); individual cases of the equation were examined earlier. D. Bernoulli (1724–1725) established that the equation (1) can be integrated by elementary functions if $ \alpha = - 2 $ or $ \alpha = - 4k( 2k- 1) $, where $ k $ is an integer. J. Liouville (1841) proved that for other values of $ \alpha $ the solution of (1) cannot be expressed in quadratures by elementary functions. The general solution of (1) can be written with the aid of cylinder functions (see [1]).

The differential equation

$$ \tag{2 } z ^ \prime = 2a( t) z + b( t) - c( t) z ^ {2} , $$

where $ a( t) $, $ b( t) $ and $ c( t) $ are continuous functions, is called a general Riccati equation (as distinct from (1), which is called a special Riccati equation). When $ c( t) \equiv 0 $, the general Riccati equation is a linear differential equation, when $ b( t) \equiv 0 $, it is a Bernoulli equation. Solutions of these two equations can be found in quadratures. Other cases of the integrability of general Riccati equations have also been studied (see [2]).

Equation (2) is closely related to the system of differential equations

$$ \tag{3 } \left . \begin{array}{c} x ^ \prime = a( t) x + b( t) y, \\ y ^ \prime = c( t) x - a( t) y. \\ \end{array} \right \} $$

If $ ( x( t), y( t)) $ is the solution of the system (3) when $ t \in I = ( t _ {0} , t _ {1} ) $ and $ y( t) $ does not vanish on $ I $, then $ z = x( t) y ^ {-} 1 ( t) $ is the solution of (2); if $ z( t) $ is the solution of (2), yet $ y( t) $ is the solution of the equation

$$ y ^ \prime = [ c( t) z( t) - a( t)] y, $$

then the pair $ ( x( t) = z( t) y( t), y( t)) $ is the solution of (3). In particular, solutions of (2) when $ c( t) \equiv 1 $ are related to solutions of the equation

$$ y ^ {\prime\prime} = 2a( t) y ^ \prime + b( t) y,\ t \in I, $$

by $ z( t) = y ^ \prime ( t) y ^ {-} 1 ( t) $ if $ y( t) $ does not vanish on $ I $. Because of this relation, the equation (2) is often drawn upon for the study of oscillation, non-oscillation (cf. Oscillating differential equation), reducibility (cf. Reducible linear system), and many other questions concerning the qualitative behaviour of linear equations and second-order systems (see [3], [4]).

The equation

$$ \tag{4 } Z ^ \prime = A( t) Z + B( t) - ZC( t) Z - ZD( t) , $$

where $ Z \equiv Z( t) $ is an unknown $ ( n \times n) $- matrix-function, and the matrix-functions $ A $, $ B $, $ C $, and $ D $ have dimensions $ n \times n $, $ n \times m $, $ m \times n $, and $ m \times m $, respectively, is called a matrix Riccati equation. The solution $ Z( t) $ of the matrix Riccati equation (4) is related to the solutions $ ( X( t), Y( t)) $ of the linear matrix system

$$ \tag{5 } \left . \begin{array}{c} {X ^ \prime = A( t) X + B( t) Y, } \\ {Y ^ \prime = C( t) X + D( t) Y, } \\ \end{array} \right \} $$

by $ X( t) = Z( t) Y( t) $.

The matrix Riccati equation plays an important part in the theory of linear Hamiltonian systems (cf. Hamiltonian system, linear), the calculus of variations, problems of optimal control, filtration, stabilization of controllable linear systems, etc. (see Control system, and [6], [7]). For example, the optimum control $ u _ {0} $ in the problem of minimization of the functional

$$ x ^ {T} ( t _ {1} ) \Phi x( t _ {1} ) + \int\limits _ { t _ {0} } ^ { {t _ 1 } } [ x ^ {T} ( t) M( t) x( t) + u ^ {T} ( t) N( t) u( t)] dt $$

on solutions of the system

$$ x ^ \prime = A( t) x + B( t) u,\ x( t _ {0} ) = x _ {0} $$

(the $ ( n \times n) $- matrices $ F $ and $ M( t) $ are symmetric and non-negative definite, and the $ ( m \times m) $- matrix $ N( t) $ is positive definite when $ t \in [ t _ {0} , t _ {1} ] $), is given by the equation

$$ u _ {0} ( t) = - N ^ {-} 1 ( t) B ^ {T} ( t) Z( t) x, $$

where $ Z( t) $ is the solution of the matrix Riccati equation

$$ \tag{6 } Z ^ \prime = - ZA( t) - A ^ {T} ( t) Z + ZB( t) N ^ {-} 1 ( t) B ^ {T} ( t) Z - M( t) $$

with the boundary condition $ Z( t _ {1} ) = \Phi $( see [5], [8]).

In problems of control on an infinite time interval, the important questions concern the existence for the matrix Riccati equation of a non-negative definite solution bounded on $ [ t _ {0} , \infty ) $, of a periodic or almost-periodic solution (in the case of periodic or almost-periodic coefficients of the equation) and the means of an approximate construction of such solutions.

In variational problems with discrete time and problems of discrete optimization the following recurrent matrix Riccati equation serves as an analogue to equation (4):

$$ Z _ {k+} 1 = A _ {k} Z _ {k} + B _ {k} - Z _ {k} C _ {k} Z _ {k} - Z _ {k} D _ {k} . $$

Equation (4) can naturally be put into correspondence with a dynamical system on a Grassmann manifold $ G _ {m,n} $( see [9]) so that the theory of dynamical systems can be applied to (4). For example, if the matrices $ A $, $ B $, $ C $, and $ D $ in (4) are periodic with period $ \omega > 0 $ and if $ | \lambda _ {1} | < \dots < | \lambda _ {n+} m | $, where $ \lambda _ {i} $ are the multipliers (see Floquet–Lyapunov theorem) of the system (5), then the corresponding dynamical system is generated by a Morse–Smale diffeomorphism (see Morse–Smale system), and is thus structurally stable.

References

[1]	J. Riccati, "Opere" , Treviso (1758)
[2]	E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1971)
[3]	N.P. Erugin, "A reader for a general course in differential equations" , Minsk (1979) (In Russian)
[4]	N.P. Erugin, "Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients" , Acad. Press (1966) (Translated from Russian)
[5]	W.T. Reid, "Riccati differential equations" , Acad. Press (1972)
[6]	R.E. Kalman, P.L. Falb, M.A. Arbib, "Topics in mathematical systems theory" , McGraw-Hill (1969)
[7]	J.-L. Lions, "Optimal control of systems governed by partial differential equations" , Springer (1971) (Translated from French)
[8]	M.K. Zakhar-Itkin, "The matrix Riccati differential equation and the semi-group of linear fractional transformations" Russian Math. Surveys , 28 : 3 (1973) pp. 89–131 Uspekhi Mat. Nauk , 28 : 3 (1973) pp. 83–120
[9]	C.R. Schneider, "Global aspects of the matrix Riccati equation" Math Syst. Theory , 7 : 3 (1973) pp. 281–286

Comments

The matrix equation $ AZ + B - ZCZ - ZD = 0 $, much related to the existence of equilibria of (4), where $ A $, $ B $, $ C $, $ D $ are given constant matrices, is known as the algebraic Riccati equation.