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

Abstract 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 ] × X 2 → 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 u ɛ ( 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 x ɛ ( t ) with x ɛ ′ ( t ) + A ( t ) x ɛ ( t ) = B ( t , x ɛ ( t ) , u ɛ ( t ) ) , t ∈ [ 0 , T ] , x ( 0 ) = 0 , to the preassigned response f ( t ) ( ∥ x ɛ - f ∥ ⩽ r ( ɛ ) ), is of the type C ɛ , 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.

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