Singular Perturbations of Nonlinear Degenerate Parabolic PDEs: a General Convergence Result

On ergodic problem for Hamilton-Jacobi-Isaacs equations

“…7 of [8] for an introduction on Hamilton-Jacobi-Bellman equations). There is a relationship between the ergodic problem and the notion of ergodicity in the Dynamical Systems Theory (see also [1][2][3]). Indeed, let us consider, for a moment, the ordinary differential equations obtained by the controlled systems (1.3), (1.4) or (1.5), where we fix the controls a(·) ∈ A and b(·) ∈ B.…”

“…We study the stronger property of uniform convergence in Ω to a constant, and this is equivalent to the ergodicity property coupled with the uniqueness of the invariant measure (see [1,2,16]). Such a dynamical system is called uniquely ergodic.…”

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

“…Introduction and preliminaries. Problems of optimal control of singularly perturbed (SP) systems have been studied intensively in both deterministic and stochastic settings (see [2] …”

“…Introduction and preliminaries. Problems of optimal control of singularly perturbed (SP) systems have been studied intensively in both deterministic and stochastic settings (see [2], [8], [17], [18], [23], [26], [28], [29], [33], [41], [52], [55], [56], [58], [61], [63], [68], [67], [65], [74], [75], [77] for a sample of the literature). Originally, the most common approaches to SP control systems, especially in the deterministic case, were related to an approximation of the slow dynamics by the solutions of the systems obtained via equating of the singular perturbations parameter to zero, with further application of the boundary layer method (see [64], [73]) for an asymptotical description of the fast dynamics.…”

“…Various averaging type approaches allowing a consideration of more general classes of SP problems, in which the optimal and near optimal controls take the form of rapidly oscillating functions and in which equating of the small parameter to zero does not lead to a right approximation, were studied in [1], [2], [6], [7], [8], [9], [10], [18], [19], [20], [27], [28], [33], [34], [38], [39], [41], [42], [44], [51], [52], [68], [67], [70], [75] (see also references therein). This research lead to a good understanding of what the "true limit" problems, optimal solutions of which approximate optimal solutions of the SP problems, are.…”