RELATIVE CONTROLLABILITY OF SEMILINEAR FRACTIONAL STOCHASTIC DELAY INTEGRODIFFERENTIAL SYSTEMS WITH DISTRIBUTED DELAYS IN THE CONTROL IN BANACH SPACES

In this research work, System (1.1), which is a class of Semilinear Fractional Stochastic Delay Integrodifferential Systems with Distributed Delays in The Control in the framework of Banach Spaces, is presented and exposed to Controllability Analysis. The principal objectives here are to State and establish necessary and sufficient conditions for Relative Controllability/ or Controllability of The System. From the results of The Controllability Analysis, the principal objectives were achieved using some Controllability Standards and the intersection property of two compact and convex set-valued functions. Advanced Mathematical tools, such as; The n×m Matrix-valued Function Definition, M^* (ς,β), The Lebesgue-Stieltjes Integration, The Unsymmetric Fubini Theorem and The Variation of Constant Formulae were made use of to cultivate The Mild Solution of The System. From the cultivated Mild Solution, key Controllability related components which our work hinges, vis a vis; “Attainable Set, Reachable Set, Target Set, Controllability Index and Controllability Grammian or Map” were carefully extracted. This work extends existing controllability theory by generalizing controllability results obtained earlier for Semilinear Fractional Stochastic Delay Integrodifferential Systems with a point Delay in the Control to a broader class involving Distributed Delays in the control.