Variational attraction of the KAM torus for the conformally symplectic system

For the conformally symplectic system $\left\{ \begin{aligned} \dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-λp, \quad λ>0 \end{aligned} \right.$ with a positive definite Hamiltonian, we discuss the variational significance of invariant Lagrangian graphs and explain how the KAM torus impacts the $W^{1,\infty}-$convergence speed of the Lax-Oleinik semigroup.

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