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, …, An−1) B(x)≠N,B(x)∈Cn(R), andB(x)(AT)j-1C(x)∈Cn-j(R) forj=1, …, n. Then\(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx0∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU−ATV, and whenever ηi(x)∈RN satisfy ηi(x0)=U(x0)di ∈ imageU(x0) η′i-Aηni(x) ∈ imageB(x),B(x)(η′i(x)-Aηi(x)) ∈Cn-1R fori=1,2.