On the feedback stabilisation of multi-time discrete delayed distributed semilinear systems

In this paper, we address the strong stabilisation and optimal control of a distributed semilinear system with multiple discrete-time delays in the state variable. These delays, commonly encountered in practical applications, pose significant challenges to the analysis and control of such systems. The stabilisation of the system is achieved by restricting its dynamics to an appropriate subspace. To this end, we propose a continuous feedback control law that ensures strong stabilisation while simultaneously minimising a specified functional, thus achieving both stability and performance optimisation. Additionally, explicit estimates of the decay rate for the stabilised state are provided, offering valuable insights into the system's transient behaviour. Various simulations and illustrative examples are presented to validate the theoretical findings and demonstrate the practical applicability of the proposed approach.