Weak Solutions to the Riccati Equation and the Application in the Closed-Loop Control

In this paper, we establish the existence of a weak solution to the Riccati equation via the canonical Hamiltonian formulation and Ekeland variational principle and present an application in the closed-loop control. It is well-known that the general Riccati equation admits no classical solution due to its blow-up behavior. Nevertheless, by introducing a residual $\mu$ through the combined application of the canonical Hamiltonian formulation and the Ekeland variational principle, we observe that under appropriate conditions, the weak solution to the Riccati equation exists. Initially, we derive the Hamilton–Jacobi equation from the Riccati equation incorporating the residual $\mu$, utilizing the canonical Hamiltonian formalism. Subsequently, we elucidate the relationship between the viscosity solution $S^{\ast }$ of the Hamilton–Jacobi equation and the residual $\mu$, thereby justifying the introduction of $\mu$ and establishing the existence of the weak solution. Finally, we present the application of the Riccati equation with the residual $\mu$ in a closed-loop control setting, thereby further substantiating the existence of the weak solution.