## On controllability of second order nonlinear impulsive differential systems.(English)Zbl 1177.34080

Semilinear, second order, infinite-dimensional control systems are considered. It is generally assumed that systems are defined in finite time interval, controls are unconstrained functions and finite number of impulses are given. First, using properties of linear unbounded operators and semigroups of linear bounded operators, a so-called mild solution is presented and its behaviour is listed. Next, definition of exact controllability in finite time interval is recalled. Using classical fixed point theorem for contractions, sufficient conditions for exact controllability in a given time interval are formulated and proved. These conditions require exact controllability of linear part of the state equation and aditional assumptions on the nonlinear part. Finally, two illustrative examples are diseussed. Moreover, many remarks and comments on the controllability for infinite-dimensional systems are presented.

### MSC:

 34H05 Control problems involving ordinary differential equations 34A37 Ordinary differential equations with impulses 34B20 Weyl theory and its generalizations for ordinary differential equations 93B05 Controllability 93C10 Nonlinear systems in control theory 93C25 Control/observation systems in abstract spaces
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