This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the \(\varepsilon \)-subdifferential of the optimal value function are derived without qualification conditions. We first calculate the \(\varepsilon \)-subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the \(\varepsilon \)-subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions.

研究了Banach空间中解集可能为空的凸优化和离散凸最优控制问题的解稳定性。对于优化问题和最优控制问题，最优值函数的\(\varepsilon \)次微分的公式都是在没有限定条件的情况下推导的。我们首先计算具有几何和函数约束的参数优化问题的最优值函数的\(\varepsilon \)次微分。然后，我们使用获得的结果推导出一个公式，用于计算具有线性状态方程、控制约束和初始、终端条件的离散凸最优控制问题的最优值函数的次微分。