Global null controllability of nonlinear delay equations with controls in a compact set

Abstract

The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM(Ω) are square integrable functions with values in the unit closed sphereL ₂([t ₀, ∞),E ^m) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t ₀,t ₁−2r], wheret ₀ is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsu∈M(Ω). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.