This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.

中文翻译：

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非线性最优控制：基于占用量度和区间分析的数值方案

本文提出了一种使用有限维线性程序和区间分析的最优控制问题的近似方案。这分为两个部分。遵循Vinter方法（SIAM J Control Optim 31（2）：518-538，1993）并使用占用度量，将最优控制问题写入无穷维线性规划问题（弱公式）。多亏了区间算法，我们通过有限维线性规划问题来解决此无限维线性规划问题。给出了有限维线性规划问题的最优值是控制问题的最优值的下界的证明。此外，根据离散化的精细程度和所选测试函数族的大小，获得的每个有限维线性规划问题的最优值形成一系列下界，这些下界收敛到初始最优控制问题的最优值。实例将说明该方法的原理。