A Study Of A Generalized Weak Kam And Aubry Mather Theory On Optimal Switching Problems

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Here, we extend the weak KAM and Aubry-Mather theories to optimal switching problems. We consider three issues: the analysis of the calculus of variations problem, the study of a generalized weak KAM theorem for solutions of weakly coupled systems of Hamilton-Jacobi equations, and the long-time behavior of time-dependent systems. We prove the existence and regularity of action minimizers, obtain necessary conditions for minimality, extend Fathi’s weak KAM theorem, and describe the asymptotic limit of the generalized Lax-Oleinik semigroup.
Overview. Dynamical systems given by Tonelli Lagrangians have been extensively studied in recent years. The deep connections between the calculus of variations, the weak KAM theory, and the
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Recently, several authors have investigated random switching problems, their weakly coupled Hamilton-Jacobi equations [1], the corresponding extensions of the weak KAM and Aubry-Mather theory [1], the long-time behavior of solutions [1], and homogenization questions [1]. In these references, as in the present paper, the state of the systems has different modes. However, in those problems, the switching between modes is driven by a random process. In contrast, here, the switches occur at deterministic times as considered [1]. Our results complement the earlier research in [1] and provide the counterpart of the aforementioned results for the optimal switching problem.
Setting of the problem. Let [1] be a compact, connected Riemannian manifold, [1] its tangent bundle, and [1] a finite set of modes. The (multimodal) Lagrangian, [1], is, for each [1], a Tonelli Lagrangian, [1], prescribing the running cost at the mode “[1]”. The switching cost is given by the function [1]. A trajectory, [1] determines both state and mode at each time. We denote by [1] the set of all absolutely continuous curves from [1] into [1], and by [1] the set of all piecewise constant functions on [1] taking values on [1]. More precisely, [1] if there exists a partition [1] of [1] such that [1]. In other words, [1] can jump in the interval [1] only at the times [1] (note that [1] is allowed to jump also at the initial and
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