Abstract
The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimensional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integro-differential system in euclidean spaces completes the discussion.
Highlights
This paper deals with the exact controllability in infinite dimensional spaces, by means of linear controls
Y (t) = Ay(t) + f (t, y(t)) + Bu(t), t ∈ [0, T ], y(t) ∈ H, with 0 < T < +∞, in the separable Hilbert space H and assume that the control term u belongs to L2([0, T ], U ) where U is a Hilbert space
System (1.1) is said to be controllable if every initial condition y0 ∈ H can be steered at time T to any y1 ∈ H, i.e. if y(0) = y0 and y(T ) = y1, by some admissible control u
Summary
This paper deals with the exact controllability in infinite dimensional spaces, by means of linear controls. The discussion in other papers is set in some Banach space, but the properties of the selection map of one control in each equivalence class are not clarified, in some cases the selection is neither introduced. As it is known, every separable Hilbert space has an orthonormal basis (en)n∈N. We assume that the Hilbert space H is compactly embedded in a Banach space (E , · E) and obtain the controllability of (1.1) by a limiting procedure This approximation solvability method was recently pointed out in [4] for the study of boundary value problems and extended in [5] to second order equations. The present controllability discussion can be extended, with suitable changes, to these models
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