On the approximate K-controllability of nonlinear systems in Banach spaces

Let X be a real Banach space. Control problems of the type (*)x′+A(t)x=B(t,x,u),t∈[0,T],x(0)=0,are considered, where, for every t∈[0,T],A(t):X⊃D(A)→X, and B:[0,T]×X2→X are given operators. The concept of approximate K-controllability (or controllability with preassigned responses) is introduced for systems of the type (*). It is shown that there exist Lipschitz-continuous approximating control functions uɛ(t),t∈[0,T], for a variety of response types. Evans-responses and Kato-responses are considered for fully nonlinear problems as well as mild ones for semilinear problems. It is also shown that the function r(ɛ), which determines the proximity of the response xɛ(t) with xɛ′(t)+A(t)xɛ(t)=B(t,xɛ(t),uɛ(t)),t∈[0,T],x(0)=0,to the preassigned response f(t) (∥xɛ-f∥⩽r(ɛ)), is of the type Cɛ, where C is a positive constant independent of ɛ. A more natural approximate K-controllability concept, “approximate eK-controllability”, is also introduced, and a result is given about it using Leray-Schauder theory. An application is given in the field of partial differential equations.