Compensation of state-dependent delays under local stabilizability assumption

In our prior work on compensation of state-dependent delays we considered globally stabilizable, forward complete systems, and designed controllers that achieve local stabilization for delay functions of the state that are arbitrary nonnegative-valued smooth functions. In this paper we remove the global assumptions and achieve the same results for plants that are locally stabilizable in the absence of delay. More specifically, we provide an affirmative answer to the question: Can nonlinear, locally stabilizable plants in the absence of input delay be stabilized by predictor-feedback in the presence of a state-dependent input delay? The key design challenge of predictor-feedback design in globally stabilizable plants is the determination of the predictor state, since the prediction horizon, that depends on the solutions of the system, is not a priori known. In the case of locally stabilizable plants, one is faced with an additional challenge: not only does the control signal have to reach the plant in finite time, but it has to reach it within the region of attraction of the delay-free plant. We resolve this challenge by providing an estimate of the time when the control signal reaches the plant. We also provide an example of a system which is neither globally stabilizable nor locally exponential stabilizable in the absence of the delay.