Impulsive second order control differential equations: Existence and approximate controllability

Abstract

The primary focus of this manuscript is on the approximate controllability of second-order semilinear control systems with impulses. There have been two sets of necessary requirements discussed. Combining the theories of the sine and cosine families, as well as the compactness of the cosine operator along with the fixed point technique (FPT) yields the first set of results. The following discussion avoids the apply of the compactness and fixed point approach of the cosine function and is shown instead using Gronwall’s inequality. The existence and uniqueness of the mild solution are also established. We have also provided a case study of theoretical outcomes that have been confirmed the theory.

Keywords

• Approximate controllability
• Second-order abstract Cauchy system
• Impulsive conditions
• Mild solution
• Gronwall’s inequality.