On the continuity property of Lp balls and an application

In this paper continuity properties of the set-valued map p→Bp(μ0), p∈(1,+∞), are considered where Bp(μ0) is the closed ball of the space Lp([t0,θ];Rm) centered at the origin with radius μ0. It is proved that the set-valued map p→Bp(μ0), p∈(1,+∞), is continuous. Applying obtained results, the attainable set of the nonlinear control system with integral constraint on the control is studied. The admissible control functions are chosen from Bp(μ0). It is shown that the attainable set of the system is continuous with respect to p.