Some remarks on the dynamics of impulsive systems in Banach spaces (Q2720151)

Here, a general class of impulsive systems in an infinite-dimensional Banach space which includes the classical model widely studied in the literature is proposed. First, the author introduces the semilinear impulsive system popularly described by the following set of evolution equations NEWLINE\[NEWLINE \dot x(t)=Ax(t)+f(x(t)), \quad t\in I\backslash D, \;x(0)=x_{0}, \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \Delta x(t_{i})=G_{i}(x(t_{i})), \;t_{i}\in D, \tag{2}NEWLINE\]NEWLINE where \(A\) is the infinitesimal generator of a \(C_{0}-\)semigroup, \(S(t), \;t\geq 0,\) in a Banach space \(E\), the functions \(f, \;G_{i}, i=1,2\ldots,n\), are continuous nonlinear maps from \(E\) to \(E\), \( \Delta x(t_{i})=x(t_{i}^{+})-x(t_{i}^{-})\) and \(D=\{t_{1}, t_{2},\ldots, t_{n}\}\in (0,T).\) NEWLINENEWLINENEWLINELet \({\mathcal B}\) denote the (complete) sigma algebra of Borel subsets of the set \(I\) and let \(\nu\) be a bounded signed measure on \({\mathcal B}\). Later, the author considers a wider class of impulsive systems that includes the model (1)--(2), namely, the evolution equation in \(E\) driven by the signed measure \(\nu\) NEWLINE\[NEWLINE dx(t)=Ax(t)dt+f(t,x(t))dt+g(t,x(t))\nu(dt), \quad t\in I:=(0,T),\tag{3}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=x_{0}.\tag{4}NEWLINE\]NEWLINE Questions of existence, uniqueness and continuous dependance of solutions on parameters such as the initial state and the measure are studied for the general model (3)--(4). Comments on some open problems conclude the paper.