Stability of functional differential equations with variable impulsive perturbations via generalized ordinary differential equations
The authors consider impulsive functional differential equations of the form \[ \begin{aligned} y'(t)&=f(t,y_t)\;\quad\text{ for } t\not=\tau_k(y(t)),\;k\in\mathbb N,\\ \Delta y(t)&=I_k(y(t))\quad\text{ for } t=\tau_k(y(t)),\;k\in\mathbb N,\end{aligned}\tag{1} \] where \(y\) and \(f\) take values in \(\mathbb R^n\), the functions \(\tau_k:\mathbb R^n\to\mathbb R\) correspond to hypersurfaces in \(\mathbb R^n\), and the impulses described by the operators \(I_k:\mathbb R^n\to\mathbb R^n\) take place whenever the solution \(y\) intersects one of the hypersurfaces. The main results of the paper are Lyapunov-type theorems related to the uniform stability and uniform asymptotic stability for the trivial solution of (1). The main idea is to convert (1) to an equivalent generalized ordinary differential equation (as introduced by \textit{J. Kurzweil}, see, e.g., [Czech. Math. J. 7(82), 418--449 (1957; Zbl 0090.30002); Generalized ordinary differential equations. Not absolutely continuous solutions. Hackensack, NJ: World Scientific (2012; Zbl 1248.34001)] and \textit{Š. Schwabik} [Generalized ordinary differential equations. Singapore: World Scientific (1992; Zbl 0781.34003)]) with values in an infinite-dimensional Banach space, and then apply the existing stability results for generalized equations. Reviewer's remark: Although the authors say that they consider equations of form (1) with state-dependent impulses, it seems that their method is applicable only to equations with preassigned moments of impulse action which are of the form \[ \begin{aligned} y'(t)&=f(t,y_t)\;\quad\text{ for } t\not=t_k^i,\;k\in\mathbb N,\;i\in\{1,\dotsc,m(\tau_k)\} \\ \Delta y(t)&=I_k(y(t))\quad \text{ for }t=t_k^i,\;k\in\mathbb N,\;i\in\{1,\dotsc,m(\tau_k)\}.\end{aligned}\tag{2} \] The reason is that, in (1), the moments \(t_k^i\) at which the solution \(y\) intersects the hypersurface given by \(\tau_k\) depend on the choice of \(y\). However, the \(t_k^i\) appear in the definition of the function \(h\) on page 199, and the existing theory of generalized differential equations requires that \(h\) does not depend on \(y\). In other words, it seems that the proofs are correct only for (2), where the moments \(t_k^i\) are the same for all solutions. Note that Lyapunov-type theorems for (2) are available in the paper \textit{M. Federson} and \textit{Š. Schwabik} [Funct. Differ. Equ. 16, No. 4, 583--607 (2009; Zbl 1200.34097)], and the results of the present article are similar to the ones there (although the assumptions are slightly different).
generalized ordinary differential equations
impulsive functional differential equations
scientific article; zbMATH DE number 6145431