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Controllability of mixed Volterra-Fredholm-type integro-differential systems in Banach space. (English) Zbl 1119.93016

Summary: M. B. Dhakne and S. D. Kendre [Commun. Appl. Nonlinear Anal. 13, No. 4, 101–112 (2006; Zbl 1119.45005)] have proved the existence of the abstract nonlinear mixed Volterra-Fredholm integro-differential system of the type

x ' (t)=ft,x(t), 0 t k(t,s,x(s))ds, 0 T h(t,s,x(s))ds,x(0)=x 0 ,tJ=[0,T]·

In this short article, we have studied sufficient conditions for controllability of semi-linear mixed Volterra-Fredholm-type integro-differential systems in Banach space of the type

x ' (t)=Ax(t)+(Bu)(t)+ft,x(t), 0 t g(t,s,x(s))ds, 0 T h(t,s,x(s))ds,x(0)=x 0 ,tJ=[0,T],

where the state x(·) takes values in a Banach space X and the control function u(·) is given in L 2 (J,U), with U as a Banach space. Here A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X. B is a bounded linear operator from U into X. The result is obtained by using the application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solutions. An example is provided to illustrate the theory.


MSC:
93B05Controllability
93C23Systems governed by functional-differential equations
45J05Integro-ordinary differential equations
45N05Abstract integral equations, integral equations in abstract spaces