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2.4. Flows on Graßmann manifolds 49
iv) If R(λm) = R(λm+1), then it follows from (2.14) and Theorems 2.4.6, 2.4.8, 2.4.9, 2.4.10, that there is no stable equilibrium of equation RDE with respect to t → ∞ and there is no solution of equation ARE whose basin of attraction (in the same sense as in (i)) is open and dense.
v) From the continuous dependence of the eigenvalues of M from the matrix coefficients, it follows that the dichotomy property cannot be spoilt by small perturbations of M. Although the system in general is not struc- turally stable, at least the dichotomic (or reverse dichotomic) solution remains attractive for almost all initial values under small perturbations of M.
vi) If the dichotomic solution Wd exists, then from (1.9), with A11 = M11, A12 = −M12, A22 = M22, we infer that
σ(M) = σ(M11 + M12Wd) ∪ σ(M22 − WdM12).
Moreover, from Theorem 2.1.2 we infer that R+ = M11 + M12Wd is the
restriction M|G(Wd). Since Im W
= Im(vm+1, . . . , vm+n) we obtain
In
d
σ(R+) = {λm+1, . . . , λm+n}. If the reverse dichotomic solution Wrd exists,
then with R− = M11 +M12Wrd we obtain similarly σ(R−) = {λ1, . . . , λn}. Hence if R(λ1 ), . . . , R(λn ) < 0, then the so-called closed loop matrix
M11 + M12Wrd
is a stable matrix.
vii) In the case of HARE, M = H is Hamiltonian then the spectrum σ(H) is symmetric with respect to the imaginary axis. Hence, a dichotomic and simultaneously a reverse dichotomic solution can only exist if there is no eigenvalue on the imaginary axis. If the reverse dichotomic solution X exists, it is the only one such that the closed loop matrix
A−SX
is stable.
Notice that instead of the possibility of a continuous extension of solutions of RDE to the Graßmann manifold Gn(Cm+n) (see the observation after (2.28)) in the case of constant coefficients we obtain solutions of RDE as meromorphic functions, hence, continuous functions in the topology induced by the chordal metric on Cˆm×n (see Remark 2.3.2 i)).
Similar and more detailed information about the behaviour of solutions of RDE can be obtained if one considers also the general case, i.e. that there are also gen- eralized eigenvectors in the Jordan basis of M. For this more detailed discussion see [AFIJ03].