Decomposition of linear singularly perturbed systems
The author considers the system of differential equations \[ (*)\quad \dot x_ 1=A_{11}x_ 1+A_{12}x_ 2+f_ 1,\quad \epsilon \dot x_ 2=A_{21}x_ 1+A_{22}x_ 2+f_ 2 \] where \(x_ i\) is a vector, \(A_{ij}=A_{ij}(t,\epsilon)\) is a matrix of appropriate dimensions, \(f_ i=f_ i(t,\epsilon)\) and \(\epsilon\) is a small parameter \((i,j=1,2)\). He gives conditions for the existence of a coordinates transformation which reduces (*) to the decoupled form \[ \dot y_ 1=A_ 1(t,\epsilon)y_ 1+f(t,\epsilon),\quad \epsilon \dot y_ 2=A_ 2(t,\epsilon)y_ 2. \] Applications to stability, control and boundary value problems are discussed.
scientific article; zbMATH DE number 4023777
scientific article