Integral manifolds and decomposition of singularly perturbed systems
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.
scientific article; zbMATH DE number 3880520
scientific article