Nonlinear stabilization in infinite dimension

Significant advances have taken place in the last few years in the development of control designs for nonlinear infinite-dimensional systems. Such systems typically take the form of nonlinear ODEs (ordinary differential equations) with delays and nonlinear PDEs (partial differential equations). In this article we review several representative but general results on nonlinear control in the infinite-dimensional setting. First we present designs for nonlinear ODEs with constant, time-varying or state-dependent input delays, which arise in numerous applications of control over networks. Second, we present a design for nonlinear ODEs with a wave (string) PDE at its input, which is motivated by the drilling dynamics in petroleum engineering. Third, we present a design for systems of (two) coupled nonlinear first-order hyperbolic PDEs, which is motivated by slugging flow dynamics in petroleum production in off-shore facilities. Our design and analysis methodologies are based on the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping. We present several simulation examples that illustrate the design methodology.