Optimal sampled-data control, and generalizations on time scales

Abstract

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of R), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.2010 Mathematics Subject Classification. 49J15, 93C57, 34N99, 39A12. 53 54 LOÏC BOURDIN AND EMMANUEL TRÉLAT R m ≤ 0,for every y ∈ Ω. In the case where kT ∈ [0, t * f ) with (k + 1)T > t * f , the above maximization condition is still valid provided (k + 1)T is replaced with t * f .

A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model

“…Concepts. The control in the force-fatigue model (1) falls into the framework of sampled-optimal control problem and we refer to [3] for more details. We use the following terminology.…”

“…A geometric analysis of the model is provided and preliminary results are presented in the framework of optimal control with sampled-data control [3]. It is based on a simplified dynamics using the geometric properties of the force-fatigue model and control reduction to simplify the physical control constraints.…”

Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model

“…Another approach being an indirect scheme based on Pontryagin type necessary conditions restricting to sampled-data control. More specifically we can refer to [12,11], adapted in [5] to deal with the FES-input and applied to the isometric case. Note that, from a more general viewpoint, sampled-data optimal control is related to compute specific (Gâteaux) types derivatives, and is in the historical continuity of the classical calculus of variations [6,20] vs Pontryagin et al maximum principle standard case, using a single type of L 1 -variations and leading to necessary optimality conditions applicable to many very practical problems [25,29].…”

“…Singular trajectories are defined and computed in the model (note that they are absent in the Ding et al model). In section 4, Pontryagin's type techniques in the permanent vs the sampled-data case, based on [12,11], are discussed in the case of Mayer problems and have to be adapted in our specific study. This leads to necessary optimality conditions in the form of Hamiltonian differential variational inequalities.…”

“…The sampled-data case motivated by digital controls is an intermediate case between calculus of variations and the general frame of Pontryagin Maximum Principle. Indeed, in [12], the authors consider L ∞ variations over a fixed interval, while time variations are considered in [11]. Due to the structure of the FES-input and the phenomenon of tetania, specific necessary Pontryagin type conditions have to be derived to deal with our muscular control problem, see [3] for the detailed presentation.…”