Abstract
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 ηi(x)∈R N satisfy ηi(x 0)=U(x 0)d i ∈ imageU(x 0) η′i-Aηni(x) ∈ imageB(x),B(x)(η′i(x)-Aηi(x)) ∈C n-1 R fori=1,2.
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Kratz, W. A limit theorem for matrix-solutions of Hamiltonian systems. Rend. Circ. Mat. Palermo 36, 457–473 (1987). https://doi.org/10.1007/BF02844900
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DOI: https://doi.org/10.1007/BF02844900