Volume 109A, number 3 PHYSICS LETTERS 13 May 1985

LINEARIZATION OF H O M O G E N E O U S RICCATI SYSTEMS

Krzysztof KOWALSKI

Department of Biophysics, Institute of Physiology and Biochemistry, Medical School of Lbdk, 3 Lindley Street, 90-131 Lbdk, Poland

Received 17 August 1984; accepted in revised form 13 March 1985

An autonomous system of n coupled Riccati equations linearizes to its linear part for arbitrary initial data.

An amazingly large number of models of natural phenomena have their statements in the form of Riccati systems [ 1 ]

Yc = c + Lx + B(x, x ) , x(0) = x0,

where L is a linear operator, L : R n ~ R n, B is a sym- metric bilinear map B: R n X R n ~ Rn;x0, c ~ R n and the dot denotes differentiation with respect to time. Nevertheless, there is not a systematic way of obtain- ing solutions of these systems so far.

In 1977 Flato et al. [2] demonstrated using meth- ods of nonlinear representations of Lie groups that the solution of the homogeneous Riccati system of the projective type [3-5]

k = L x + ( a ' x ) x , x ( 0 ) = x 0, (1)

for n = 2 can be [educed to the solution of its linear part. Recently, observations of Flato et al. [2] have been extended by Reed [6] to the case with arbitrary n.

The main advantage of the Flato-Reid lineariza- tion algorithm is that it presents in a systematic manner an exphcit and invertible map connecting the solution of the nonlinear system (1) and the solution of its lin- ear part for almost all initial conditions. Unfortunately, this linearization technique has also some flaws. Name- ly, (i) as mentioned above, it cannot be applied for some initial data x 0 (the analycity of the linearization map is violated for some initial conditions x0), (ii) it cannot be applied for a singular matrix L.

The aim of this letter is to show that the system (1) can be easily solved for arbitrary initial data x 0 and

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for arbitrary matrix L. Consider the system (1). We seek the solution

x ( x O, t) of the system (1) in the form

x(x o, t)= a(t)A(Xo, t), (2)

where ct(0) = 1 and A (x O, t) designates the solution of the linear system

.~i = L A , A(O) = x O. (3)

One can easily check that the presented procedure of the solution of the system (1)is equivalent to the well- known method of variations of constants. Substituting (2) into (1) the following one-dimensional equation is obtained

&=ct2a-A, a ( 0 ) = l ,

Solving this elementary equation and using eq. (2) we get

x(x 0, t) = A(x O, t) (4a) 1 -f[~a.A(xo, r)dr

a(x o, t) (4b)

1 - L - l a - [A(x0, t) - x 0 l '

where eq. (4b) holds for reversible L and L designates the transpose of the operator L. It thus appears that the homogeneous Riccati system of the projective type (1) linearizes to its linear part for arbitrary initial data x 0 and for arbitrary matrix L.

Now we introduce an alternative method which also

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Volume 109A, number 3 PHYSICS LETTERS 13 May 1985

solves the general initial value problem (1) in a system- atic manner. This method was suggested by a referee. Consider the system (1). The transformation

x = B/(1 - a" B) (5)

(where/~(0) = x 0, B(0) = 0) is the linearization of the system (1) to the following homogeneous, second- order one

B = LB. (6)

The transformation (5) can be easily deduced from the one-dimensional counterpart of the system (1). It follows immediately from (6), (5) and (3) that the solution of the system (1) is given by eq. (4a).

The Fla to-Reid linearization map connecting the solution o f the system (1) and the solution o f its line- ar part can be obtained from eq. (4b) from dividing the numerator and the denominator by the factor F,

F = 1 + L - l a . x o .

This map reduces the solution o f the system (1) to the solution of the linear system (3) with the initial data A(O) = xo/F. These initial data are only nonsingular for such x 0 that F :# 0 (the inequality F ~ 0 is pre- cisely the condition required for the Fla to-Reid lin- earization map to be well defined).

I would like to thank a referee for helpful comments.

References

[1] E.H. Kerner, J. Math. Phys. 22 (1981) 1366. [2] M. Flato, G. Pinczon and J. Simon, Ann. Sci. Ec. Norm.

Sup. 10 (1977) 405. [3] R.L. Anderson, Lett. Math. Phys. 4 (1980) 1. [4] R,L. Anderson, J. Harnard and P. Winternitz, Lett. Math.

Phys. 5 (1981) 143. [5] R.L. Anderson, J. Harnard and P. Winternitz, Physica 4D

(1982) 164. [6] J.L. Reid, Phys. Lett. 97A (1983) 309.

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