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. In this paper the simpler problem in the optimal design of tension structures is considered. Let a square inextensible network undergo given transverse loads and be prestressed in its plane by means of tractions applied along the boundary. We want to study the optimal distribution of tractions in order to minimize the compliance of the net when the total pretraction load and two positive lower and upper bounds to the pretractions are given. Attention is addressed to general issues; we discuss existence and uniqueness and provide a description of certain qualitative properties of the solution. The compliance of the net is defined as the work done by the loads evaluated in the equilibrium configuration. It is then a functional of the pretractions and these, in their turn, are the coefficients of an equilibrium operator. Thus, the problem reduces to studying the minimization of a cost functional defined over a certain class of operators, as is customary in many other optimization problems [1-12]. The theory of (/-convergence (see, e.g., [13, 14]) proves to be the natural setting to discuss the existence of solutions for these problems. At this point, however, because of the constancy of the pretractions along the fibers (cf. [15]) G-convergence is in the peculiar circumstance of being equivalent to weak convergence, and the topological properties that are required for a solution to exist can be assessed in a more familiar context. The situation is close to that encountered in dealing with the one-dimensional problems where G-convergence is equivalent to weak convergence of the inverses of the operator coefficients, [16, 17], so that one could in fact obtain existence results without mentioning G-convergence. Velte and Villaggio [18], for instance, follow this approach in a paper which raised our interest in the present problem. This distinguished feature of G-convergence in one dimension, however, does not carry over to a higher dimension. So, the fact that a property of this sort holds in the present case gives the problem a special location in structural optimization. After proving the existence of optimal pretractions, we study other features of the solutions. In particular, by making use of Kuhn-Tucker conditions, it is possible to show that all solutions correspond to the same configuration of the net and that the

A problem in the optimal design of networks under transverse loading