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In this paper we discuss the controllability of two-point boundary valueproblem (TBVP) for one-dimensional wave equation. Some new concepts areintroduced: TBVP input control problem, minimum-input solution (MS) andpre-minimum-input solution (PMS). We set the metric in $L^1$ and $L^2$ spaceson a closed set, and control the input to reach its minimum. And we mainlydiscuss the property of input, the existence and uniqueness of MS and PMS for$L^1$ and $L^2$ metric respectively. The minimum inputs lie on a strip in $L^1$and PMS for $L^1$ and $L^2$ always exists. Furthermore, to construct PMS, wealso introduce an approximation method which meets certain conditions.

Controllability of two-point boundary value problem for wave equations in $L^1$ and $L^2$ spaces: One dimensional case