Adaptive stabilization of LTI systems with distributed input delay

We solve stabilization problems of LTI systems with unknown parameters and distributed input delay. The key challenge is that the infinite-dimensional input dynamics are distributed, which makes traditional infinite-dimensional backstepping inapplicable. We resolve this challenge by employing backstepping-forwarding transformations of the finite-dimensional state of the plant and of the infinite-dimensional actuator state, which enable us to design Lyapunov-based update laws. Using the estimations of the unknown parameters, the certainty equivalent linear state feedback results in a nonlinear adaptive control law. We illustrate our design with a simulation example.