Figalli, Gomes and Marcon (2016) extended weak KAM theory to deterministic optimal switching problems: for a finite family of Tonelli Lagrangians on a compact manifold and a switching cost satisfying a strict triangle-type inequality, they built the calculus of variations of the multimodal action (existence and regularity of minimizers, Euler-Lagrange equations with momentum jumps at switch times), proved a generalized weak KAM theorem (critical constant, Peierls barrier, projected Aubry set, calibrated curves), and established uniform convergence of the generalized Lax-Oleinik semigroup to a critical solution. Three layers of the classical theory, however, were not developed in this switching setting: invariant Mather-type probability measures minimizing the averaged action including switching costs, the structural characterization of the projected Aubry set, and quantitative rates for the large-time convergence. The problem is to construct this missing Aubry-Mather theory for deterministic optimal switching: the measure-theoretic objects, their duality with the critical constant and critical subsolutions, the structure of the Aubry set, and the rates (or sharp obstructions) governing the convergence.

Background

Weak KAM theory, initiated by Fathi, connects the long-time behavior of first-order Hamilton-Jacobi equations with the dynamics of the associated variational problem. For a single Tonelli Lagrangian L:TM→RL:TM\to\mathbb{R} -- a C2C^2 function on the tangent bundle of a compact connected Riemannian manifold MM that is convex and superlinear in the velocity variable -- let HH be its Legendre dual Hamiltonian. The evolution equation ∂tu+H(x,Du)=0\partial_tu+H(x,Du)=0 is solved, in the viscosity-solution sense (the standard weak solution notion for fully nonlinear first-order equations), by the Lax-Oleinik semigroup Ttu0T_tu_0, which shifts the initial datum by the minimal action accumulated along curves. Fathi's weak KAM theorem states that Ttu0+c0tT_tu_0+c_0t converges uniformly to a solution of the stationary equation H(x,Du)=c0H(x,Du)=c_0, where the critical value c0c_0 is the unique constant for which subsolutions exist. The limit is selected by the Peierls barrier, the liminf over large times of the minimal action plus c0tc_0t, and by the Aubry set, the set of points where this barrier vanishes, which is the locus of recurrent minimizing behavior and is traversed by calibrated curves, i.e. curves that realize equality in the subsolution inequalities against every critical subsolution. The measure-theoretic counterpart, going back to Mather, studies invariant probability measures on TMTM that minimize the averaged action ∫L dμ\int L\,d\mu: their existence by compactness, their duality with critical subsolutions (the minimum averaged action equals −c0-c_0), and the containment of their supports in the Aubry set.

For systems of Hamilton-Jacobi equations, two distinct coupling mechanisms must be distinguished. Standard weakly coupled systems have the form ∂tui+Hi(x,Dui)+∑jdijuj=0\partial_tu_i+H_i(x,Du_i)+\sum_jd_{ij}u_j=0 with monotone coupling coefficients, which corresponds to randomly switching between modes; for this class the analogues of the whole weak KAM / Aubry-Mather program are by now developed: large-time convergence (Camilli-Ley-Loreti-Nguyen; Mitake-Tran), Aubry sets (Davini-Zavidovique), a cycle characterization of the Aubry set (Ibrahim-Siconolfi-Zabad), generalized Mather measures (Terai), and Lagrangian representations via random frames and random or twisted Lax-Oleinik semigroups (Mitake-Siconolfi-Tran-Yamada; Davini-Siconolfi-Zavidovique; Zavidovique). Optimal switching problems are structurally different: the controller deterministically chooses when to switch between modes and pays a cost ψ(x,i,j)\psi(x,i,j) for each switch, a classical control setting (Capuzzo Dolcetta-Evans) whose stationary problem is the obstacle-coupled, quasi-variational system

max⁡{H(x,Du(⋅,i),i)−c0, u(⋅,i)−Ψu(⋅,i)}=0,Ψu(⋅,i)=min⁡j≠i{u(⋅,j)+ψ(⋅,i,j)};\max\{H(x,Du(\cdot,i),i)-c_0,\ u(\cdot,i)-\Psi u(\cdot,i)\}=0,\qquad \Psi u(\cdot,i)=\min_{j\neq i}\{u(\cdot,j)+\psi(\cdot,i,j)\};

Gomes-Serra studied such implicit-obstacle systems by PDE methods.

Figalli-Gomes-Marcon built the weak KAM theory for deterministic optimal switching. On M×IM\times\mathcal{I} with I={1,…,m}\mathcal{I}=\{1,\dots,m\} a finite mode set, Tonelli Lagrangians L(⋅,⋅,i)L(\cdot,\cdot,i) and a switching cost ψ\psi (continuous, ψ(⋅,i,i)≡0\psi(\cdot,i,i)\equiv0, satisfying a strict triangle-type inequality, i.e. a strict form of the triangle inequality ensuring that switching from mode ii to mode jj through an intermediate mode is strictly more expensive than switching directly, which excludes costless chattering between modes, with C2C^2 regularity used in part of the analysis), they considered the generalized action of curves γ=(γM,γI)\gamma=(\gamma_M,\gamma_{\mathcal{I}}) -- an absolutely continuous path in MM together with a piecewise-constant mode path -- given by the integral of L(⋅,⋅,i)L(\cdot,\cdot,i) plus the switching costs at the jump times of the mode path. They proved existence and regularity of minimizers, Euler-Lagrange equations between switches with momentum jump conditions at switch times, conservation of the energy, semiconcavity (a one-sided second-order regularity property) of the minimal action hth_t, a weak KAM theorem with a critical constant c0c_0, construction of the Peierls barrier and of the projected Aubry set A⊂M×I\mathcal{A}\subset M\times\mathcal{I}, existence of calibrated curves through every Aubry point, and the uniform convergence Ttu0→vT_tu_0\to v of the generalized Lax-Oleinik semigroup to a critical solution represented via the barrier and A\mathcal{A}. What the paper does not provide is the measure-theoretic and quantitative layer: no invariant (Mather-type) probability measures minimizing the averaged multimodal action are constructed, no structural characterization of A\mathcal{A} is given, and the convergence theorem carries no rate.

Problem Statement

Develop the missing Aubry-Mather layer of the weak KAM theory for deterministic optimal switching, in the setting of Figalli-Gomes-Marcon: compact connected Riemannian manifold MM, finite mode set I={1,…,m}\mathcal{I}=\{1,\dots,m\}, Tonelli Lagrangians L(⋅,⋅,i)L(\cdot,\cdot,i), and switching cost ψ\psi that is continuous, satisfies ψ(⋅,i,i)≡0\psi(\cdot,i,i)\equiv0 and the strict triangle-type inequality, with the C2C^2 regularity used in part of the source analysis. Three jointly required results form one objective, the analogue for deterministic switching of the classical Aubry-Mather chapter:

(a) Measures. Define an appropriate class of invariant probability measures for the multimodal dynamics -- measure-theoretic objects encoding trajectories that follow the Euler-Lagrange flow of L(⋅,⋅,i)L(\cdot,\cdot,i) while in mode ii and pay the switching costs ψ\psi at switch times -- prove that minimizers of the averaged generalized action exist in this class, and establish the duality with the critical constant c0c_0 and with critical subsolutions: the infimum of the averaged action over the class equals −c0-c_0, and the supports of minimizing measures are characterized through the projected Aubry set A\mathcal{A}.

(b) Structure. Characterize the structure of A⊂M×I\mathcal{A}\subset M\times\mathcal{I}: its decomposition into dynamically invariant components and the role of switching cycles of modes, in analogy with the cycle characterizations known for standard weakly coupled systems.

(c) Rates. Determine the quantitative large-time behavior of the convergence Ttu0→vT_tu_0\to v from the source's large-time theorem: dynamical conditions (on the dynamics restricted to A\mathcal{A} and on switching cycles) under which the convergence is exponential or algebraic of a given order, together with examples showing the rates are sharp.

Part (a) is the central object, (b) its support structure, and (c) its quantitative consequence for the semigroup, mirroring the way these three layers form one theory in the classical case. The switching costs ψ\psi must be treated as part of the action: an answer that reduces the problem to a standard weakly coupled system without switching costs does not resolve it. Extensions to weaker assumptions on ψ\psi (non-strict triangle inequality or weaker regularity) and to infinite, state-dependent, or hybrid deterministic-stochastic mode sets are recorded as separate neighboring directions and are not part of this problem.

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