Convex Quadratic Equation

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.