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Asymptotic expansion of the solution of a singularly perturbed matrix Riccati equation with an infinitely large initial condition. (English. Russian original) Zbl 0805.93026

J. Comput. Syst. Sci. Int. 30, No. 6, 24-30 (1992); translation from Izv. Ross. Akad. Nauk, Tekh. Kibern. 1992, No. 1, 83-89 (1992).
Summary: Under certain conditions, the asymptotic expansion of the solution of an equation of the form \[ (A'+ \varepsilon B')dK (t,\varepsilon)/dt= - K'(t,\varepsilon) C(t)- C'(t) K(t,\varepsilon)+K'(t,\varepsilon) S(t) K(t,\varepsilon)- W(t) \] with given initial condition \((A'+ \varepsilon B')K(T,\varepsilon)= F(\varepsilon)\) is found by constructing the asymptotic expansions of the solution of three auxiliary problems. Here, \(t\in [0,T]\), \(\varepsilon> 0\) is a small parameter, the matrix \(A\) is singular, \(A+\varepsilon B\) is invertible for sufficiently small \(\varepsilon\neq 0\), and the prime denotes the transpose. In addition to functions of \(t\), the asymptotic expansions contain boundary-layer functions \(V_ j(\tau)\) satisfying the estimate \(\| V_ j(\tau)\|\leq c\exp(\kappa\tau)\), where \(c\), \(\kappa> 0\), \(\tau= (t- T)/\varepsilon^ p\) and \(p\) is determined by the properties of the matrices \(A\) and \(B\).

MSC:

93C15 Control/observation systems governed by ordinary differential equations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
93C73 Perturbations in control/observation systems
34A45 Theoretical approximation of solutions to ordinary differential equations
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