Nonnegative control of finite-dimensional linear systems |
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| Authors: | Jérôme Lohéac Emmanuel Trélat Enrique Zuazua |
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| Institution: | 1. Université de Lorraine, CNRS, CRAN, F-54000 Nancy, France;2. Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France;3. Chair in Applied Analysis, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universiät Erlangen-Nürnberg, 91058 Erlangen, Germany;4. Chair of Computational Mathematics, Fundación Deusto, University of Deusto, 48007 Bilbao, Basque Country, Spain;5. Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain;1. IMPA, Estrada Dona Castorina 110, Rio de Janeiro 22460-320, RJ, Brazil;2. IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas-SP, Brazil;3. Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal;1. Department of Mathematics, Temple University, Philadelphia, PA 19122, United States of America;2. Instituto Argentino de Matemática A. P. Calderón, CONICET, Buenos Aires, Argentina;1. Laboratoire IMATH, Université de Toulon, BP 20132, 83957 La Garde Cedex, France;2. Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy;3. Dipartimento di Matematica e Informatica, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy |
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| Abstract: | We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations. |
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| Keywords: | Minimal time Nonnegative control Dirac impulse Temps minimal Contrôles positifs Impulsions de Dirac |
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