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Necessary conditions for the junction of interior arcs and boundary arcs of an optimal control are given. These conditions are based on necessary conditions for the switching function holding at junction points or contact points with the state boundary. The junction theorems obtained are similar to junction theorems in singular control problems. I. Formulation of the problem We consider the following control problem with control appearing linearly: determine the scalar, piecewise continuousdoi:10.1137/0315023 fatcat:wjvgidx2xfafxoxlz6imhxflsm