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**Approximate controllability of nonlinear deterministic and stochastic systems with unbounded delay.**
*(English)*
Zbl 1220.93011

Summary: We consider approximate controllability for nonlinear deterministic and stochastic systems with resolvent operators and unbounded delay. We study the problem of approximate controllability of deterministic nonlinear differential equations with impulsive terms, resolvent operators and unbounded delay. Next, approximate controllability results are being established for a class of nonlinear stochastic differential equations with resolvent operators in a real separable Hilbert space. By using the resolvent operators and fixed point technique, sufficient conditions have been formulated and proved. In this paper, we prove the approximate controllability of nonlinear deterministic and stochastic control systems under the assumption that the corresponding linear system is approximately controllable. Examples are presented to illustrate the utility and applicability of the proposed method.

### MSC:

93B05 | Controllability |

93E03 | Stochastic systems in control theory (general) |

93E20 | Optimal stochastic control |