Determination of the parameter of the differential equation of fractional order with the Caputo derivative in Hilbert space

Abstract

We consider an abstract differential equation of the first order with unbounded linear operator Dαu(t) = Au(t) + f(t) in a Hilbert space H, where Dα is the Caputo fractional derivative. For this equation the Cauchy problem is studied with initial data u(0) = u0 at t = 0. We assume that the operator A is self-adjoint and non-positive. The inverse problem to determine the nonhomogeneous member is considered under the assumption that this term has the following representation f(t) = φ(t)p, where the scalar function φ(t) is given and the element p is an unknown element of the space H. This problem belongs to the class of inverse problems. Inverse problems for abstract differential equations was discussed initially with this inhomogeneous structure member but for equations with an operator generating a C0-semigroup and usual derivative. An additional condition u(T ) = u1 is specified for the determination of the unknown p, where u1 is a given element of the space H. Thus we get the two-point problem which is only beginning to be studied for the case of fractional derivatives. The question of existence and uniqueness of the classical solution of the inverse problem is studied. Sufficient conditions of correct solvability of the inverse problem are obtained. Explicit formula to determine the unknown element in the differential equation is given.