Well-posedness of Lur'e systems with feedthrough

For a large class of Lur'e systems with time-varying nonlinearities and feedthrough we consider several well-posedness issues, namely: existence, continuation, blow-up in finite-time, forward completeness and uniqueness of solutions. Lur'e systems with feedthrough are systems of forced, nonlinear ordinary differential equations coupled with a nonlinear algebraic equation determining the output of the system. The presence of feedthrough means that the algebraic equation is implicit in the output, and, in general, the output may not be expressible by an analytic formula in terms of the state and the input. Simple examples illustrate that the well-posedness properties of such systems are not necessarily guaranteed by assumptions sufficient for the corresponding well-posedness properties of Lur'e systems without feedthrough. We provide sufficient conditions for the well-posedness properties mentioned above, using global inversion theorems from real analysis and tools from non-smooth analysis and differential inclusions. The theory is illustrated with examples.