×

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.

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
49J22 Optimal control problems with integral equations (existence) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agrarwall RP, Nonlinear Analysis 40 pp 1– (2000)
[2] Ahmed NU, Pitman Research Notes in Maths Series 246, in: Semigroup Theory with Applications to System and Control (1991)
[3] DOI: 10.1016/S0362-546X(03)00117-2 · Zbl 1030.34056
[4] Yan-Lai Chen, Acta Analysis Functionalis Applicata 5 pp 322– (2003)
[5] Guo D, Journal of Mathematical Analysis and Applications 177 pp 538– (1993) · Zbl 0787.45008
[6] Hu S, Handbook of Multivalued Analysis (Theory) (1997)
[7] Zhimin He, Journal of Mathematical Analysis and Applications 296 pp 8– (2004) · Zbl 1057.45002
[8] Lakshmikantham V, Theory of Impulsive Differential Equations (1989)
[9] Xunjing Li, Optimal Control Theory for Infinite Dimensional Systems (1995) · Zbl 0817.49001
[10] Liu X, The Canadian Applied Mathematics Quarterly 3 pp 419– (1995)
[11] Peng Yunfei, Guizhou Science 20 pp 30– (2002)
[12] Xiang X, Journal of Guizhou University 4 pp 335– (2003)
[13] Xiang Xiaoling, Acta Mathematices Applicates Sinica 16 pp 27– (2000) · Zbl 1005.49017
[14] Xiang X, Discrete and Continuous Dynamical Systems pp 911– (2005)
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.