Controllability of nonlinear neutral evolution integrodifferential systems
The authors consider the functional integrodifferential control system \[ {d\over dt}[x(t)+g(t,x_t)]=A(t)x(t)+\int^t_0 B(t,s)x(s) ds+(Gu)(t)+f\left(t,x_t,\int^t_0h(t,s,x_s) ds\right), \] \[ x_0=\varphi, \] where \(t\in J=[0,b], x(t)\) belongs to the Banach space \(X, u(\cdot)\in L^2(J,U), U\) is a Banach space, \(A(t)\) and \(B(t,s)\) are closed linear operators on \(X\) with dense domain \(D(A)\) which is independent of \(t,\) \(G\colon U\to X\) is a bounded linear operator, \(h\colon J\times J\times C([-r,0],X)\to X\), \(f\colon J\times C([-r,0],X)\times X\to X\) and \(g\colon J\times C([-r,0],X)\to X\) are continuous functions, \(\varphi\in C([-r,0],X), x_t(\theta)= x(t+\theta)\) for \(\theta\in[-r,0]\). Sufficient conditions for controllability of the above system are established. The main result is obtained by applying the Schaefer fixed-point theorem. An application to partial integrodifferential equations is also given.
functional integrodifferential control system
scientific article; zbMATH DE number 1866692
scientific article