In this paper, we investigate an L-0 optimization problem with constraints in a form of Volterra integral equation and the L-infinity- norm. In particular, the equivalence theorem among the L-p optimizations with p is an element of [0.1] is derived, which provides the following twofold extension of the existing results: First, these theoretical results enable us to solve sparse optimal control problems without imposing the finite dimensionality of the system to be controlled, which was the crucial assumption for the derivation of the existing results. Second. the relationship between the partial state constrained problem and output controllability is newly characterized.