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…show more content… Recently, several authors have investigated random switching problems, their weakly coupled Hamilton-Jacobi equations , the corresponding extensions of the weak KAM and Aubry-Mather theory , the long-time behavior of solutions , and homogenization questions . 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 . Our results complement the earlier research in  and provide the counterpart of the aforementioned results for the optimal switching problem.
Setting of the problem. Let  be a compact, connected Riemannian manifold,  its tangent bundle, and  a finite set of modes. The (multimodal) Lagrangian, , is, for each , a Tonelli Lagrangian, , prescribing the running cost at the mode “”. The switching cost is given by the function . A trajectory,  determines both state and mode at each time. We denote by  the set of all absolutely continuous curves from  into , and by  the set of all piecewise constant functions on  taking values on . More precisely,  if there exists a partition  of  such that . In other words,  can jump in the interval  only at the times  (note that  is allowed to jump also at the initial and