Rothe’s Fixed Point Theorem and the Controllability of the Benjamin-Bona-Mahony Equation with Impulses and Delay

For many control systems in real life, impulses and delays are intrinsic phenomena that do not modify their controllability. So we conjecture that under certain conditions the abrupt changes and delays as perturbations of a system do not destroy its controllability. There are many practical examples of impulsive control systems with delays, such as a chemical reactor system, a financial system with two state variables, the amount of money in a market and the savings rate of a central bank, and the growth of a population diffusing throughout its habitat modeled by a reaction-diffusion equation. In this paper we apply the Rothe’s Fixed Point Theorem to prove the interior approximate controllability of the following Benjamin-Bona-Mahony (BBM) type equation with impulses and delay ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) 1 , , , , , , in 0, , , 0, on 0, , , , , ,0 , , , , , , , , , 1, 2,3, , , t t k k k k k k z a z b z u t x f t z t r x u t x z t x z s x s x s r x z t x z t x I t z t x u t x k p ω τ τ φ + −  − ∆ − ∆ = + − ×Ω  = ×∂Ω   = ∈ − ∈Ω   = + =   where 0 a ≥ and 0 b > are constants, Ω is a domain in N  , ω is an open nonempty subset of Ω , 1ω denotes the characteristic function of the set ω , the distributed control ( ) ( ) 2 0, ; u C L τ ∈ Ω , [ ] : ,0 r φ − ×Ω→  are continuous functions and the nonlinear functions [ ] , : 0, k f I τ × × →    are smooth enough functions satisfying some additional conditions.