A class of interior-point trust-region algorithms for infinite-dimensional nonlinear optimization subject to pointwise bounds in L-p-Banach spaces, 2 less than or equal to p less than or equal to infinity, is formulated and analyzed. The problem formulation is motivated by optimal control problems with L-p-controls and pointwise control constraints. The interior-point trust-region algorithms are generalizations of those recently introduced by Coleman and Li [SIAM J. Optim., 6 (1996), pp. 418-445] for finite-dimensional problems. Many of the generalizations derived in this paper are also important in the finite-dimensional context. All first- and second-order global convergence results known for trust-region methods in the finite-dimensional setting are extended to the infinite-dimensional framework of this paper.