Approximate Controllability Of Fractional Differential Inclusions With Nonlocal Conditions

In 2005,N.Heymans and I.Podlubny introduced that non-local initial value problems a4e more practical in describing physical phenomena than general initial value problems.Compared with integer-order differential equations,fractional-order differential equations have better advantages and can describe the properties of objects and reflect objective facts more accurately.The differential inclusion system studied in this paper is an approximate con-trollability problem of fractional-order differential inclusion system with non-local initial value conditions.In existing articles,it is generally assumed that non-local items are fully continuous or global,and it is obviously not easy to prove in many cases.Therefore,we weaken the non-local term condition and assume that non-local term satisfies the local increase condition.Using fractional derivatives and integrals,semigroup theory and Bohnenblust-Karlin fixed point theorem prove that there is a weak solution in the differential inclusion system,and give reasonable assumptions.Finally,the sufficient conditions for the approximate controllability of the system are obtained.