On hybrid systems (Q2719504)

The authors consider a hybrid system of the form NEWLINE\[NEWLINE \begin{aligned} &\frac{dx}{dt} = f(t,x,y),\quad x(0) = x_0,\tag{1}\\ &y(t) = y_0 + \int\limits_0^\tau g(t,s,x(s),y(s)) ds,\tag{2} \end{aligned} NEWLINE\]NEWLINE with \( x\in \mathbb{R}^k\), \( y\in \mathbb{R}^l\), the functions \(f\) and \(g\) are definite and continuous for \( t\in\Delta\), \( s\in\Delta\) \( (\Delta = [0,\tau])\), \( x\in \mathbb{R}^k\), \( y\in \mathbb{R}^l\), and satisfy Lipschitz conditions. Existence and uniqueness conditions are established for the solution to system (1), (2) under different assumptions on the functions \(f\) and~\(g\).