Approximate controllability for Riemann-Liouville fractional differential equations

Abstract

The article objectifies the approximate controllability of fractional nonlinear differential equations having Riemann-Liouville derivatives. The nonlinear term involved in the equation, also depending on the control parameter u(·), is considered to be locally Lipschitz. First, the existence of solutions is deduced using Lipschitz condition, semigroup theory and fixed point approach. Then sufficient condition for approximate controllability of the system is established using Cauchy convergence through iterative and approximate techniques. The theory of semigroup together with probability density function has been uti- lized to reach the desired conclusions. Lastly, an application is provided to support the proposed methodology.