Nonlinear impulsive integro-differential equations of mixed type and optimal controls. (English) Zbl 1101.45002

The authors study the existence of a PWC-mild solution for impulsive integro-differential equations of mixed type: \[ \dot x(t) +Ax(t)=F(t,x(t), (G\dot x)(t), (Sx)(t)) ,\quad t\in(0,T)\backslash D, \]
\[ x(0)= x_{0},\quad \Delta x(t_{i})= J_{i}(x(t_{i})),\quad i=1,\dots,n, \] where \(D=\{ t_{1},\dots,t_{n}\} \subset(0,T) ,\) \(G\) and \(S\) are given nonlinear integral operators, and \(-A\) is the infinitesimal generator of a \(C_{0}\)-semigroup on an infinite dimensional Banach space. Next, an existence result of optimal controls for a Lagrange problem is proved. An example illustrates the theoretical results.


45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
49J22 Optimal control problems with integral equations (existence) (MSC2000)
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