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Carleman commutator approach in logarithmic convexity for parabolic equations
September 2018, 8(3&4): 935-964. doi: 10.3934/mcrf.2018041
Controllability under positivity constraints of semilinear heat equations
Dario Pighin 1,, and Enrique Zuazua 1,2,3,4,
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
DeustoTech, Fundación Deusto, Avda. Universidades, 24, 48007, Bilbao, Basque Country, Spain
Facultad de Ingeniería, Universidad de Deusto, Avda. Universidades, 24, 48007, Bilbao, Basque Country, Spain
Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 06, CNRSUMR 7598, Sorbonne Universités, F-75005, Paris, France
* Corresponding author: Dario Pighin
Dedicated to Professor Jiongmin Yong on the occasion of his 60th birthday
Received November 2017 Revised April 2018 Published September 2018
Fund Project: This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, FA9550-14-1-0214 of the EOARD-AFOSR, the MTM2014-52347 and MTM2017 Grants of the MINECO (Spain) and ICON of the French ANR.
In many practical applications of control theory some constraints on the state and/or on the control need to be imposed.
In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive.
More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory.
We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.
Keywords: Semilinear heat equations, controllability under constraints, iterative method, dissipativity, waiting time.
Mathematics Subject Classification: Primary: 35k58, 35k51; Secondary: 49J99.
Citation: Dario Pighin, Enrique Zuazua. Controllability under positivity constraints of semilinear heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 935-964. doi: 10.3934/mcrf.2018041
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Figure 1. Stepwise procedure
Figure 2. Illustration of the proof of Theorem 1.3 in two steps: Stabilization + Control
Figure 3. Final data for the adjoint system
Figure 4. Evolution of the adjoint heat equation with final datum $\varphi_T$
Figure 5. graph of the control in the minimal time
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Dario Pighin Enrique Zuazua