Investigation of invariant sets of impulsive systems by Lyapunov functions (Q2703284)

By means of Lyapunov function the authors study the positive invariant and local invariant sets of the following system of differential equations with impulsive action: NEWLINE\[NEWLINEdx/ dt=X(t,x), \quad t\neq \tau_{i}(x), \quad \Delta x|_{t=\tau_{i}(x)}=I_{i}(x), \quad \tau_{i}(x)<\tau_{i+1}(x),NEWLINE\]NEWLINE where \(X(t,x)\), \(I_{i}(x)\) are continuous and Lipschitz in \(x\in D\), \(D\subset {\mathbb{R}}^{n}\). Let \(V(t,x)\geq 0\) be a continuously differentiable function on \(t\geq 0\), \(x\in \overline D_{1}\subset D\) and let \(\text{Proj}_{\mathbb{R}^{n}}N_{t}\) be compact in \(D_{1}\), where \(N_{t}\) is a null set of function \(V(t,x)\). If NEWLINE\[NEWLINE{\partial V(t,x)\over\partial t}+\sum_{i=1}^{n} {\partial V(t,x)\over\partial x_{i}}X_{i}(t,x)\leq 0, \quad V(\tau_{i}(x),x+I_{i}(x))\leq V(\tau_{i}(x),x), \quad t\geq 0, x\in D_{1},NEWLINE\]NEWLINE then the set \(V(t,x)=0\), \(x\in D_{1}\), \(t\geq 0\), is a positive invariant set for the given system. Sufficient conditions for the stability of the positive invariant sets are obtained.