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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