Then, for a sufficiently large number in modulus [lambda] [member of] [[PHI].sub.[??],[psi]], the
inverse operator [(q - [lambda]I).sup.-1] exists and is continuous in space H = [L.sup.2](0, 1), and the following estimate holds:
Thus, we have proved that the operator T has an
inverse operator [T.sup.-1].
The main strategy of the method of operator is to find the
inverse operator of the primary differential problem, i.e.
REDUCING THE NUMBER OF
INVERSE OPERATORS IN THE CARTESIAN RULES
By finding the
inverse operator and imposing initial condition we obtain
Therefore, by the
inverse operator theorem, we obtain that [[bar.T].sup.1] is a bounded linear operator.
Although their relationship was linked to the general IMC strategy, they apply equally well to the neural network inverse model based IMC strategy here, where the
inverse operator or controller is approximated by the neural network inverse model instead.
where the notation A-1 denotes the
inverse operator for the linear operator A of (18).
In view of (4) and (17), by applying the
inverse operator [I.sup.[alpha]] on both sides of (16) and solving corresponding integrals we get
Generalized
Inverse Operators: And Fredholm Boundary-Value Problems, 2nd Edition