RATE OF STABILITY OF SOLUTIONS OF MATRIX POLYNOMIAL AND QUADRATIC EQUATIONS

In this paper the rate of stability of solutions of matrix polynomial equations of the type A(0)+A(1)X+A(2)X(2)+...+A(m)X(m) = 0 is studied. Particular attention is given to the case where the matrix polynomial L(lambda):=A(0)+A(1) lambda+A(2) lambda(2)+...+A(m) lambda(m) is weak ly hyperbolic, i.e., for every non-zero vector a the scalar polynomial [L(lambda)x, x] has only real roots. Also the rate of stability of so lutions of matrix quadratic equations of the type XBX+XA-DX-C=0 is stu died. Here the special case that is of interest to continuous-time opt imal control theory, that is, the case where B=B is positive semidefi nite and C=C, A=-D*, is discussed in detail. The analogous theory for the discrete-time optimal control leads to the equation