On the approximate image-controllability of nonlinear systems in Banach spaces Original Research Article

Let XX be a real Banach space. Control problems of the type equation(*) View the MathML sourcex′+A(t)x=B(t,x,u),t∈[0,T],x(0)=0, Turn MathJax on are considered, where, for every View the MathML sourcet∈[0,T],A(t):X⊃D(A)→X, and B:[0,T]×X2→XB:[0,T]×X2→X are given operators. The concept of approximate KK-controllability (or controllability with preassigned responses) is introduced for systems of the type (*).(*). It is shown that there exist Lipschitz-continuous approximating control functions View the MathML sourceuɛ(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),r(ɛ), which determines the proximity of the response xɛ(t)xɛ(t) with View the MathML sourcexɛ′(t)+A(t)xɛ(t)=B(t,xɛ(t),uɛ(t)),t∈[0,T],x(0)=0, Turn MathJax on to the preassigned response f(t)f(t) (∥xɛ-f∥⩽r(ɛ)∥xɛ-f∥⩽r(ɛ)), is of the type Cɛ,Cɛ, where CC is a positive constant independent of ɛɛ. A more natural approximate KK-controllability concept, “approximate eKeK-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.