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As a complement to two recent papers by Toan and Yao (Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J Global Optim. 2014;58:595–612), and by An and Toan (Differential stability of convex discrete optimal control problems. Acta Math Vietnam. 2018;43:201–217) on subdifferentials of the optimal value function of discrete optimal control problems, this paper studies the differential stability of convex discrete optimal control problems under control constraints, where the solution set may be empty. By using a suitable sum rule for ϵ-subdifferentials and a suitable product rule for ϵ-normal directions, we obtain formulas for computing the ϵ-subdifferential of the optimal value function. Several illustrative examples are also given. [ABSTRACT FROM AUTHOR]