##
**Nonlinear impulsive systems on infinite dimensional spaces.**
*(English)*
Zbl 1030.34056

Summary: The authors consider two different classes of nonlinear impulsive systems one driven purely by Dirac measures at a fixed set of points and the second driven by signed measures. The later class is easily extended to systems driven by general vector measures. The principal nonlinear operator is monotone hemicontinuous and coercive with respect to certain triple of Banach spaces called Gelfand triple. The other nonlinear operators are more regular, nonmonotone continuous operators with respect to suitable Banach spaces.

We present here a new result on the compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.

We present here a new result on the compact embedding of the space of vector-valued functions of bounded variation and then use this result to prove two new results on existence and regularity properties of solutions for impulsive systems described above. The new embedding result covers the well-known embedding result due to Aubin.

### MSC:

34G20 | Nonlinear differential equations in abstract spaces |

34K45 | Functional-differential equations with impulses |

34A37 | Ordinary differential equations with impulses |

46G10 | Vector-valued measures and integration |

47J35 | Nonlinear evolution equations |

### Keywords:

nonlinear; impulsive; systems; infinite-dimensional spaces; signed measures; vector measures; embedding; nonlinear impulsive systems; Dirac measures; general vector measures; Gelfand triple
PDFBibTeX
XMLCite

\textit{N. U. Ahmed} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 54, No. 5, 907--925 (2003; Zbl 1030.34056)

Full Text:
DOI

### References:

[1] | Ahmed, N. U., Compactness in certain abstract function spaces with applications to differential inclusions, Discuss. Math. Differential Inclusions, 15, 21-28 (1995) · Zbl 0828.46003 |

[2] | Ahmed, N. U., Vector measures for optimal control of impulsive systems in Banach spaces, Nonlinear Funct. Anal. Appl., 5, 2, 96-106 (2000) · Zbl 0982.49022 |

[3] | Ahmed, N. U., Some remarks on the dynamics of impulsive systems in banach spaces, Dyn. Continuous Discrete Impulsive Systems, 8, 261-274 (2001) · Zbl 0995.34050 |

[4] | Ahmed, N. U., Systems governed by impulsive differential inclusions on Hilbert spaces, Nonlinear Anal., 45, 693-706 (2001) · Zbl 0995.34053 |

[5] | Ahmed, N. U., State dependent vector measures as feedback controls for impulsive systems in Banach spaces, Dyn. Continuous Discrete Impulsive Systems, 8, 251-261 (2001) · Zbl 0990.34056 |

[6] | Ahmed, N. U., Measure solutions impulsive evolutions differential inclusions and optimal control, Nonlinear Anal., 47, 13-23 (2001) · Zbl 1042.49505 |

[7] | Ahmed, N. U., Necessary conditions of optimality for impulsive systems on Banach spaces, Nonlinear Anal., 51, 409-424 (2002) · Zbl 1095.49513 |

[8] | Ahmed, N. U., Nonstandard impulsive evolution equations in Banach spaces, Nonlinear Funct. Anal. Appl., 7, 3, 437-453 (2002) · Zbl 1032.34058 |

[9] | Ahmed, N. U.; Teo, K. L., Optimal Control of Distributed Parameter Systems (1981), North-Holland: North-Holland New York, Oxford · Zbl 0472.49001 |

[10] | Lakshmikantham, V.; Bainov, D. D.; Simenov, P. S., Theory of Impulsive Differential Equations (1999), World Scientific: World Scientific Singapore, London |

[11] | X. Li, J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Systems and Control: Foundations and Applications, Birkhauser, Boston, Basel, Berlin, 1995.; X. Li, J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Systems and Control: Foundations and Applications, Birkhauser, Boston, Basel, Berlin, 1995. |

[12] | Liu, J. H., Nonlinear impulsive evolution equations, Dyn. Continuous Discrete Impulsive Systems, 6, 1, 77-85 (1999) · Zbl 0932.34067 |

[13] | Pandit, S. G.; Deo, S. G., Differential systems involving impulses, Lecture Notes in Mathematics, Vol. 954 (1982), Springer: Springer Berlin, Heidelberg, New York · Zbl 0417.34085 |

[14] | E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. IIA, IIB, Springer, New York, 1986.; E. Zeidler, Nonlinear Functional Analysis and its Applications, Vols. IIA, IIB, Springer, New York, 1986. · Zbl 0583.47050 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.