Decomposition of systems with discontinuous controls whose elements operate at different speeds (Q1114920)

The authors consider the dynamical system \[ (*)\quad \mu {\dot \eta}=A\eta +B\sigma +Cs;\quad \mu {\dot \sigma}=F\eta +G\sigma +Ls,\quad \dot s=\sigma;\quad \dot y=Dy+\phi (z,s,y,t) \] where \(\mu\) is a small positive parameter; \(t\in [t_ 0,\infty)\); \(\eta \in R^ m\); \(u,\sigma,s\in R^ 1\); \(y\in R^ n\); \(z=(u,\sigma)^ T\); \(\phi\) is of C-class; A,B,C,F,G,L,b are matrices depending continuously on a small parameter; D is a constant matrix. The coordinates of the control function are constrained: \(u_ k(s)=\beta_ k+sgn(s_ k)\), \(| \beta_ k| \leq 1\), \(k=1,...,l\). A special property of (*) is that sliding can occur iff \(\sigma_ k=s_ k=0\), \(k=1,...,l.\) The paper deals with the decomposition of (*). A new formula describing the system dynamics enables to analyse the stability of the system.