ABSTRACT
In this section we deal in a Banach space X with a closed linear operator A with a dense domain. For the purposes of the present section we have occasion to use two mappings~: [0, T] 1-+ £(X) and F: [a, T] 1-+ X and fix two arbitrary elements Uo, U1 EX. The· two-point inverse problem consists of finding a function u E C1 ([0, ~J1; X) and an element p E X from the set of relations
(7.1.1)
(7.1.2)
(7.1.3)
(7.1.4)
u'(t) = A u(t) + /(t) , u(O) = Uo, /(t) = ~(t) P+ F(t) , u(T) = U1'
One assumes, in addition, that the operator A is the generator of a strongly continuous semigroup V(t) or, what amounts to the same things, the
Cauchy (direct) problem (7.1.1)-(7.1.2) is uniformly well-posed. It is known from Fattorini (1983) that for Uo E "D(A) and
/ Ee1 ([0, 1']; X) +e([o, 1']; "D(A») the direct problem (7.1.1)-(7.1.2) has a solution in the class of functions
Au E e([O, 1']; X) . This solution is unique in the indicated class of functions and is representable by
In contrast to the direct problem the inverse problem (7.1.1)-(7.1.4) involves the function / as the unknown of the prescribed structure (7.1.3), where the element p E X is unknown and the mappings ~ and Fare available. Additional information about the function u in the form of the final overdetermination (7.1.4) provides a possibility of determining the element p.
