Abstract
We study the controllability of a coupled system of linear parabolic equations, with nonnegativity constraint on the state. We establish two results of controllability to trajectories in large time: one for diagonal diffusion matrices with an “approximate” nonnegativity constraint, and a another stronger one, with “exact” nonnegativity constraint, when all the diffusion coefficients are equal and the eigenvalues of the coupling matrix have nonnegative real part. The proofs are based on a “staircase” method. Finally, we show that state-constrained controllability admits a positive minimal time, even with weaker unilateral constraint on the state.
Highlights
In the following, N and N∗ denote the sets of respectively nonnegative and positive integers
We show that there exists a positive minimal controllability time as soon as the initial state and the target trajectory are different, even if we allow the state to be greater than a negative constant instead of being nonnegative (Thm. 2.12)
We have studied the problem of nonnegative controllability for coupled reaction-diffusion systems
Summary
N and N∗ denote the sets of respectively nonnegative and positive integers. State-constrained controllability is a challenging subject that has gained popularity in the last few years, notably at the instigation of Jerome Loheac, Emmanuel Trelat and Enrique Zuazua in the seminal paper [16], in which some controllability results with positivity constraints on the state or the control for the linear heat equation are proved, under a minimal time condition which turns out to be necessary This question yielded to an increasing number of articles in different frameworks, many of them being coauthored by Enrique Zuazua: for ODE systems [17, 18], semilinear and quasilinear heat equations [22, 24], monostable and bistable reaction-diffusion equations [20, 26], the fractional one-dimensional heat equation [4, 7], wave equations [25], and age-structured systems [19].
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