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The goal of this paper is to study an output stabilization problem: the gradient stabilization for linear distributed systems. Firstly, we give definitions and properties of the gradient stability. Then we characterize controls which stabilize the gradient of the state. We also give the stabilizing control which minimizes a performance given cost. The obtained results are illustrated by simulations in the case of one-dimensional distributed systems.

The aim of this paper is to study regional stabilization of the flux of bilinear distributed systems. More precisely it consists in studying the asymptotic behavior of the gradient of such a system not in its whole geometrical evolution domain Ω but only in a subregion ω of Ω. Then we give definitions and under suitable condition we give gradient stabilizing control. We also characterize the control which stabilizes regionally the gradient, and minimizes a given performance cost. Then we develop a numerical approach that is successfully illustrated by simulations.

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