ILC method for solving approximate controllability of fractional differential equations with noninstantaneous impulses

In this paper, we utilize fractional calculus, theory of semigroup and fixed point approach to prove existence and approximate controllability results for a class of fractional differential equations (FDEs) with noninstantaneous impulses by constructing a suitable composite control function, imposing that the associated linear problem is approximately controllable on the terminal subinterval and dividing our global task into many subtasks on each subinterval. Next, we apply P-type iterative learning control (ILC) updating law to generate a sequence of control functions to find a desired control function to guarantee the error between the output and the desired reference trajectories tending to zero via a suitable norm in the sense of uniform convergence. As a result, the limit of the sequence of control functions is the required control to guarantee the above problem is approximately controllable. In addition, a numerical example is illustrated to demonstrate ILC scheme to solve approximate controllability of time fractional impulsive PDEs via tracking the given continuous and discontinuous trajectory.