Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is re-examined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton–Jacobi equation, Hamilton–Jacobi inequality, and Hamilton–Jacobi–Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

两个主要结果（A）和（B）以代数封闭形式表示。（A）关于凸二次方程，这是首次在文献中和在统一的框架中制定了一个解析的等价可溶条件和所有解的参数化。该哲学基于矩阵代数，同时通过新颖的等价/坐标变换（相对于秩不足的Hessian矩阵更具挑战性的情况）得到了促进。另外，验证了参数解双射。从（A）的角度出发，重新考虑了一个主要应用，该应用说明了另一个主要结果（B），该结果涉及无限和有限时间范围非线性最优控制。借助（A），与汉密尔顿–雅各比方程相关的底层凸二次方程，分别明确地解决了汉密尔顿-雅各比不等式和汉密尔顿-雅各比-贝尔曼方程。因此，可以在每个解决方案集中捕获对最优控制器组成的长期追求，即关联值函数的梯度。展望未来，为剩余的对称条件准备了使用状态相关的（分别为微分的）Riccati方程方案精确定位最优性的初步方法。