Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 284, Issue 10, Pages -Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2023.109855
Keywords
Bilinear optimal control; Shape optimisation; Bang -bang property
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We investigate the bang-bang property for L infinity - L1 constrained bilinear optimal control problems in one-dimensional torus with parabolic equations. Our study is motivated by applications in applied mathematics, especially in reaction-diffusion models. We prove that if the functions j1 and j2 are increasing, then the maximizer m* of the functional is bang-bang, represented by m* = 1E for some subset E of the torus. This result also establishes an existence property for shape optimization problem.
We investigate the bang-bang property for fairly general classes of L infinity - L1 constrained bilinear optimal control problems in the case of parabolic equations set in the one-dimensional torus. Such a study is motivated by several applications in applied mathematics, most importantly in the study of reaction-diffusion models. The main equation writes partial differential tum - partial differential 2xxum = mum + f(t, x, um), where m = m(x) is the control, which must satisfy some L infinity bounds (0 5 m 5 1 a.e.) and an L1 constraint (' m = m0 is fixed), and where f is a non-linearity that must only satisfy that any solution of this equation is positive at any given time. The functionals we seek to optimise are rather general; they write ,7 (m) = similar to (0,T )xT j1(t, x, um) + ' T j2(x, um(T, center dot)). Roughly speaking we prove in this article that, if j1 and j2 are increasing, then any maximiser m* of ,7 is bang-bang in the sense that it writes m* = 1E for some subset E of the torus. It should be noted that such a result rewrites as an existence property for a shape optimisation problem. Our proof relies on second order optimality conditions, combined with a fine study of two-scale asymptotic expansions. In the conclusion of this article, we offer several possible generalisations of our results to more involved situations (for instance for controls of the form m phi(um) or for some time-dependent controls), and we discuss the limits of our methods by explaining which difficulties may arise in other settings. (c) 2023 Elsevier Inc. All rights reserved.
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