标题： 动态优化及其与经典和量子约束系统的关系 显示英文标题

标题： Dynamic optimization and its relation to classical and quantum constrained systems

摘要： 我们从物理学家的角度研究由一个状态变量和一个控制变量组成的简单动态优化问题的结构。 通过类比物理模型，我们在经典和量子框架下研究这个系统。 在经典情况下，动态优化问题等价于一个受约束的经典力学系统，因此我们必须使用狄拉克方法以正确的方式对其进行分析。 我们发现该模型中有两个第二类约束：一个固定与控制变量相关的动量，另一个是最优控制定律的提醒。 该约束系统的动态演化由规范变量与哈密顿量的狄拉克括号给出。 这种动态结果与由庞特里亚金方程给出的无约束情况完全相同，这些方程是我们的物理优化问题的正确经典运动方程。 在相同的庞特里亚金方案中，通过施加闭环$\lambda$-策略，作用量的最优性条件给出一个一致性关系，这与动态规划方法的哈密顿-雅可比-贝尔曼方程相关。 通过量化经典模型，也得到了类似的结果。 通过在量子薛定谔方程中设置波函数$\Psi(x,t) = e^{iS(x,t)}$，得到关于$S$函数的非线性偏微分方程。 对于右边的量化，当$S(x,t)$被识别为最优值函数时，这就是哈密顿-雅可比-贝尔曼方程。 因此，贝尔曼最大原理中的哈密顿-雅可比-贝尔曼方程可以被解释为优化问题的量子方法。 显示英文摘要

摘要： We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a classical mechanics constrained system, so we must use the Dirac method to analyze it in a correct way. We find that there are two second-class constraints in the model: one fix the momenta associated with the control variables, and the other is a reminder of the optimal control law. The dynamic evolution of this constrained system is given by the Dirac's bracket of the canonical variables with the Hamiltonian. This dynamic results to be identical to the unconstrained one given by the Pontryagin equations, which are the correct classical equations of motion for our physical optimization problem. In the same Pontryagin scheme, by imposing a closed-loop $\lambda$-strategy, the optimality condition for the action gives a consistency relation, which is associated to the Hamilton-Jacobi-Bellman equation of the dynamic programming method. A similar result is achieved by quantizing the classical model. By setting the wave function $\Psi(x,t) = e^{iS(x,t)}$ in the quantum Schr\"odinger equation, a non-linear partial equation is obtained for the $S$ function. For the right-hand side quantization, this is the Hamilton-Jacobi-Bellman equation, when $S(x,t)$ is identified with the optimal value function. Thus, the Hamilton-Jacobi-Bellman equation in Bellman's maximum principle, can be interpreted as the quantum approach of the optimization problem.