Abstract
The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multi-point condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.
Highlights
This paper concerns the second order integro-differential equation ztt(t, x) = ∆z(t, x) + f t, x, z(t, x), h(x, ξ)z(t, ξ) dξ + b(x)v(t, x), (1)D for x ∈ D ⊂ Rn bounded and with a sufficiently regular boundary and t ∈ [0, T ] := J which is a model for the description of diffusive behaviours such as the propagation of electro-magnetic waves, the motion of a string or a membrane with external damping, the evolution of visco-elastic fluids and the heat propagation
The main aim of this paper is to investigate the possibility to act by v(t, x) for obtaining a solution of (1) which satisfies some given nonlocal condition (for instance (3) or (5)) and reaches a prescribed configuration at time t = T, i.e. z(T, ·) = z1(·), where z1(·) stands for a suitable real valued function
The main results of the paper are Theorems 3.1 and 3.2; they provide sufficient conditions for the controllability of (8)-(9)
Summary
The main aim of this paper is to investigate the possibility to act by v(t, x) for obtaining a solution of (1) which satisfies some given nonlocal condition (for instance (3) or (5)) and reaches a prescribed configuration at time t = T , i.e. z(T, ·) = z1(·), where z1(·) stands for a suitable real valued function. This is known as the exact controllability problem associated to (1) (for short controllability problem in the sequel).
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