Continuing former work ,  the authors consider a mechanical system that models a segment of a live or artificial worm or a baloon for angioplasty that is placed within a cylindrical compliant tube (vein). The statics of the inflation process is based on the Principle of Minimal Potential Energy. This is handled as an optimal control problem with state constraint. Certain peculiarities make the necessary optimality conditions go beyond those from classical textbooks. A careful analysis of the conditions leads to a boundary value problem describing the shape of the inflated system and to the determination of the contact forces between balloon and vein. Simulation results are to be presented in a forthcoming Part 2.