Exponential stabilization and minimization problem for a delayed semilinear system

Abstract

In this study, a weaker condition is introduced to address the exponential stabilization of semilinear systems in Hilbert spaces. Compared to the entire state space, verification is made easier by this condition, which is defined on a subspace of the phase space and connected to the system's initial condition. Interestingly, the condition is easier to verify as the first condition's norm gets closer to zero. Furthermore, the method avoids the requirement to explicitly find the semigroup's form, which is frequently difficult in reality. Additionally, a minimization problem is taken into account, and it is proven that the optimal control exists and is unique, guaranteeing exponential stabilization. Numerical simulations are used to validate the theoretical results through two examples.