Antistable Wave PDE with Input Delay

In this chapter we continue with the designs for delay-PDE cascades as in Chapter 18. Here we deal with an antistable wave PDE, which has all of its infinitely many eigenvalues in the right half-plane (all located on a vertical line).

The wave PDE problem with input delay is much more complex than the delayheat cascade in Chapter 18. The primary reason is the second-order-in-time character of the wave equation, though the “antistability” of the plant also creates a challenge.

Due to the extra complexity of the wave PDE, in this chapter we forego the derivation of explicit closed-loop solutions such as those that we derived in Section 18.8. However, we do derive the explicit expressions for the control gains and present a stability analysis.

We present the design for an antistable wave PDE with input delay in Section 19.1 and explain its origins in the baseline delay for an antistable wave PDE without delay in Section 19.2. The explicit solutions for the controller’s gain kernels are derived in Section 19.3. The stability analysis is presented in two steps, first for the target system in Section 19.4, and then for the system in the original variables in Section 19.5.