By Dettmar J.

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4_ (Ell _ Q12,~21 ))Zl0 = z20" The same type of consistency condition is obtained for T < co with X replaced by X ( t ) . g. e. -B21u(t) is just a constant matrix multiplied to zl(t). 7) for the case that E is singular, we can perform the transformations that transform 44 c~E - ]~A in Kronecker canonical form also with this equation. 7). 39) 0 = Q22 - o,~21R - 1 S ,2x. 39) is a consistency condition. 40) Y21(t) = ( I - C ,-,21R - i B *21]~-l(_Q21 4- $21R -1 Sll• + S21R-1B~IYII(Q). 37) to yield a standard Riccati equation for Yll(t).

17 EXAMPLE. 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The Kronecker canonical form of this pencil is as follows: '0 "0 0 1 1 0 I 1 0 0 OE 0 1 0 0 0 1 0 1 0 { 0 0 1 0 0 0 1 0 1 Clearly there is no pairing for the mentioned types of blocks. We now give the corresponding results for the discrete case. 18 PROPOSITION. Let (ce, 13) e C, (a, fl) ~ (0,0) and let z(J) E C 2"+m, j = 0 , . . ,k such that .

5) as the optimal feedback solution. 6) 0= C*QC+E*XA+A*XE-(B*XE+S*C)*R-I(B*XE+S*C). So it is not surprising that deflating subspaces will also be used to obtain solutions to algebraic Riecati equations, independently of any control problems in the background. g. Lancaster/Rodman [L 3]. 7 THEOREM. 6). c)] ' span an n-dimensional subspace of C 2"+m , which is a deflating subspace for a A - 1313. 8). 6) in the second component. For B we obtain BV= - E = - X E. Thus, the columns of V span a deflating subspace for ~ A - fiB.