A limit theorem for matrix-solutions of Hamiltonian systems

The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A ^n?1) B(x)≠N,B(x)∈C _n(R), andB(x)(A ^T)^j-1 C(x)∈C _n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx ₀∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A ^TV, and whenever η_i(x)∈R ^N satisfy η_i(x ₀)=U(x ₀)d _i ∈ imageU(x ₀) η′_i-Aη_ni(x) ∈ imageB(x),B(x)(η′_i(x)-Aη_i(x)) ∈C _n-1 R fori=1,2.