As has been shown in the literature (Rothe and Rothe, 2010; Rothe and Scholtz, 2003), the description of a mechanical system in terms of canonical transformations together with the Hamilton–Jacobi equation for the $S$ function is ill-defined when the system has second class constraints. In this case, Carathéodory’s integrability conditions are violated and either the corresponding Hamilton–Jacobi equation cannot be solved or their solutions do not describe the system at all. This can be remedied by enlarging the phase space so that the constraints become first class in the extended space. Another way to approach this problem, is to apply the Rothe–Scholtz method discussed in Rothe and Rothe (2010) and Rothe and Scholtz (2003), so that the constraints themselves become variables of a new canonical transformation. This method works when the elements of the Dirac matrix are constant. On the other hand, it has been shown that optimal control theory can be written in phase space as a mechanical system with second class restrictions (Itami, 2001; Hojman, 0000; Contreras et al., 2017; Contreras and Peña 2018). This implies that the description of control theory can become inconsistent in terms of the Hamilton–Jacobi equation. In this article we will use the description of Rothe–Scholtz to analyse a subclass of LQ linear-quadratic problems whose Dirac matrix is constant and to check if the integrability conditions can be fulfilled so as to not get inconsistencies.

正如文献所显示的那样（Rothe和Rothe，2010； Rothe和Scholtz，2003），根据正则变换以及哈密顿-雅各比方程对机械系统的描述。 $\mathrm{小号}$当系统具有第二类约束时，函数定义不明确。在这种情况下，违反了Carathéodory的可积性条件，或者相应的Hamilton-Jacobi方程无法求解，或者它们的求解根本无法描述该系统。这可以通过扩大相空间来解决，使约束成为扩展空间中的头等舱。解决该问题的另一种方法是应用Rothe和Rothe（2010）以及Rothe和Scholtz（2003）中讨论的Rothe-Scholtz方法，以便约束本身成为新规范转换的变量。当Dirac矩阵的元素恒定时，此方法有效。另一方面，已经证明，最优控制理论可以写在相空间中，作为具有第二类限制的机械系统（Itami，2001； Hojman，0000； Imtra，2001）。Contreras等人，2017年; Contreras andPeña2018）。这意味着就汉密尔顿-雅各比方程而言，控制理论的描述可能会变得不一致。在本文中，我们将使用Rothe–Scholtz的描述来分析Dirac矩阵恒定的LQ线性二次问题的子类，并检查是否可以满足可积性条件，以免出现不一致的情况。