On exact controllability of semilinear systems

Abstract

In this paper, we suggest a proof method for the exact controllability of semilinear systems. Previously, it was applied in the context of the approximate controllability. In comparison to the traditional proof methods, the suggested method is easy because it does not require heavy estimations. Also, it is effective because it does not need some unnatural conditions, which are required for application of fixed-point theorems. Using these advantages of the suggested method, we prove that a semilinear system in a Hilbert space, which satisfies general conditions for the existence of its solution, steers any initial state to any final state from the domain of the system operator if its linear part is exactly controllable at every noninitial time moment, the nonlinear term is bounded, and the norm of the inverse of the Gramian increases as time goes to zero not faster than the reciprocal function. The obtained result is demonstrated on examples.