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Integral manifolds and decomposition of singularly perturbed systems. (English) Zbl 0552.93017

The paper uses the method of integral manifolds to find a transformation under which the singularly perturbed system of differential equations \[ \dot x=f(t,x,y,\epsilon),\quad \epsilon \dot y=g(t,x,y,\epsilon) \] reduces to \[ \dot u=F(t,u,\epsilon),\quad \epsilon \dot v=G(t,u,v,\epsilon). \] The assumptions are similar to those in Tichonov’s theorem, i.e. the eigenvalues of the matrix \((\partial g/\delta y)\{t,x,h_ 0(t,x),0)\), where \(h_ 0(t,x)\) is the isolated solution of \(g(t,x,y,0)=0\), have strictly negative real parts. A transformation with the above properties is given for linear systems. As an application, the linear boundary value problem that arises in linear- quadratic optimal control is considered. The transformation helps to obtain an asymptotic expansion of the solution.
Reviewer: A.Dontchev

MSC:

93B17 Transformations
34E15 Singular perturbations for ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
93C10 Nonlinear systems in control theory
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