On the stability of invariant sets of discontinuous dynamical systems (Q2768822)

The authors study an impulsive system of the form NEWLINE\[NEWLINE\dot{x}=f(x),\quad x\not\in \Gamma,\quad \Delta x|_{x\in \Gamma}=Tx-x:=I(x),\tag{1}NEWLINE\]NEWLINE with \(x\in \overline{D}\subset \mathbb{R}^n\), \(f(x)\in C(\overline{D} \mapsto \mathbb{R}^n)\), \(D\) is a bounded domain in \(\mathbb{R}^n\), \(t\in [0,\infty)\), \(T\in C(\Gamma \mapsto Q)\), \(\Gamma \subset \overline{D}\) is a hyper-surface transversal to the vector field \(f(x)\), \(Q\) is a closed subset of \(\overline{D}\). It is assumed that any solution \(x(t,x_0)\) satisfying the condition \(x(0,x_0)=x_0\) exists on \([0,\infty)\).NEWLINENEWLINENEWLINEBasing on ideas of the direct Lyapunov method, the authors establish sufficient conditions for existence, stability, asymptotic stability and instability of invariant sets for (1). Let, e.g., there exists a function \(V(x)\in C(\overline{D} \mapsto \mathbb{R})\) and a set \(M_0\subset\overline{D}\), \(M_0\neq\emptyset,\) which satisfy the following conditions NEWLINE\[NEWLINEV(x)=0\quad \forall x\in M_0, \qquad V(x)>0\quad \forall x\in \overline{D}\setminus M_0,NEWLINE\]NEWLINE and let \(K_0\) be the class of continuous functions \(\varphi :\mathbb{R}_+ \mapsto \mathbb{R}_+\) such that \(\varphi(s)=0\Leftrightarrow s=0\). One of the results obtained by the authors is as follows: if \(\langle \text{grad} V(x),f(x)\rangle \leq-\varphi(V(x))\), \(x\not\in \Gamma\), for appropriate \(\varphi(s)\in K_0\) then \(M_0\) is an asymptotically stable invariant set for (1).NEWLINENEWLINENEWLINEThe authors also study stability properties of invariant sets with respect to solutions which start on a given subset of the phase space.