Existence, Stability and Optimality of a Semilinear Control System Involving Instantaneous and Non-instantaneous Impulses

This article deals with the existence, stability and optimality of a semilinear control systems with instantaneous and non-instantaneous impulses. We study this problem by using the technique of mild solution and fixed point theory together with theory of semigroups. In the field of applied sciences impulses are of great significance for a dynamical systems, as mathematical models and dynamical systems representing any real life scenario it is not necessary to have continuous dynamics. This problem where both impulses are discussed together is opening the door for many researhers to work and generalize this concept for stochastic, fractional and delay cases by considering the qualitative properties as well. One example is also given in the last to show the applicability of the problem considered in the main system.