Characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems

Abstract

The aim of this paper is to study singularly perturbed control systems. Firstly, we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics. Secondly, we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system.

Reflected dynamics: Viscosity analysis for L∞ cost, relaxation and abstract dynamic programming

“…The dynamics is seen as occupation measures satisfying convenient constraints given by the state restrictions K and the differential formula. First, the main difficulty and novelty with respect to the abundant existing literature (see for instance [8,7,28,15]) is the presence of of the reflection elements z ∈ N K (x). Therefore, a further component is added to the occupation measures.…”

“…Of course, this is a relaxation technique and one must ensure that the associated value function is the same. For usual (non-reflected) dynamics, this makes the object of [14] (see also [8,7,28,15]). The constraints appearing in the linear formulation deal with the differential formula for regular test functions and they can also capture state-constraints for the dynamics.…”

On Near Optimal Control of Systems with Slow Observables

“…Being rewritten in the "shrinked" time scale t = ǫτ , such systems are commonly called weakly coupled singularly perturbed (SP) systems. Problems of control of SP systems (including weakly coupled SP systems) have received a great deal of attention (see [2], [3], [4], [5], [6], [11], [12], [13], [14], [17], [22], [23], [33], [35], [36], [39], [40], [43], [45], [46], [48], [50], [51] and references therein). However, to the best of our knowledge, in all the earlier works, except the recent paper [45], the SP control systems were considered under the assumption that the set of occupational measures generated by the controls and the corresponding solutions of the reduced system dy(τ ) dτ = f 1 (0, u(τ ), y(τ )) (8) is independent of the initial conditions from a sufficiently large set (the validity of this being guaranteed by imposing certain stability or controllability type conditions).…”

Some support considerations in the asymptotic optimality of two-scale controlled PDMP

“…This is very much inspired by the literature on singularly perturbed systems (e.g. [2][3][4][5][6][7] ). This kind of singular-perturbation problem is particularly challenging because, in many cases, one fails to identify the limit dynamics.…”