Singular Optimal Control of a 1-D Parabolic-Hyperbolic Degenerate Equation
Gueye, Mamadou; Lissy, Pierre (2016), Singular Optimal Control of a 1-D Parabolic-Hyperbolic Degenerate Equation, ESAIM: Control, Optimisation and Calculus of Variations, 22, 4, p. 1184 - 1203. 10.1051/cocv/2016036
TypeArticle accepté pour publication ou publié
Journal nameESAIM: Control, Optimisation and Calculus of Variations
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Abstract (EN)In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove that the cost of the control explodes exponentially fast in small time and converges exponentially fast in large time in some adapted weighted norm. The main tools used are a spectral decomposition involving Bessel functions and their zeros, some usual results on admissibility of scalar controls for diagonal semigroups, and the moment method of Fattorini and Russell. Keywords: degenerate parabolic equations, cost of the control, uniform controllability, degenerate transport equation.
Subjects / KeywordsDegenerate parabolic equation; cost of the control; uniform controllability; Bessel functions
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