The function space , , allows for functions with jump discontinuities and is thus attractive for treating optimal control problems with discrete-valued control functions. We show that while arbitrary chattering controls are impossible, there exist feasible controls in that have countably jump discontinuities with jump height one in each of countably many pairwise disjoint intervals. However, under mild assumptions, we show that certain types of jump discontinuities cannot be optimal. The derivation of meaningful optimality conditions via a direct variational argument using simple feasible perturbations remains a major challenge; as illustrated by an example.
L’espace fonctionnel , , est compatible avec les discontinuités et est par conséquent un candidat de choix pour résoudre des problèmes de contrôle optimal avec des fonctions de contrôle à valeurs discrètes. Nous montrons que, bien que les contrôles fortement oscillants soient impossibles, il existe des contrôles admissibles dans ayant un nombre fini de discontinuités avec un saut de 1 pour chacune des paires dénombrables d’intervalles disjoints. Cependant, sous des hypothèses raisonnables, nous montrons que certaines de ces discontinuités ne peuvent pas être optimaux. Établir des conditions d’optimalité pertinentes via un argument variationnel avec des perturbations admissibles simples constitue un défi majeur, ce que nous illustrons par un exemple.
Revised:
Accepted:
Published online:
Paul Manns 1; Thomas M. Surowiec 2
@article{CRMATH_2023__361_G9_1531_0, author = {Paul Manns and Thomas M. Surowiec}, title = {On {Binary} {Optimal} {Control} in $H^s(0,T)$, $s < 1/2$}, journal = {Comptes Rendus. Math\'ematique}, pages = {1531--1540}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.507}, language = {en}, }
Paul Manns; Thomas M. Surowiec. On Binary Optimal Control in $H^s(0,T)$, $s < 1/2$. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1531-1540. doi : 10.5802/crmath.507. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.507/
[1] Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. II. Existence theorems for weak solutions, Trans. Am. Math. Soc., Volume 124 (1966) no. 3, pp. 413-430 | DOI | MR | Zbl
[2] Wavelet-based approximations of pointwise bound constraints in Lebesgue and Sobolev spaces, IMA J. Numer. Anal., Volume 42 (2020) no. 1, pp. 417-439 | DOI | MR | Zbl
[3] Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | DOI | MR | Zbl
[4] On sliding optimal states, Dokl. Akad. Nauk SSSR, Volume 143 (1962) no. 6, pp. 1243-1245 | MR
[5] Partial outer convexification for traffic light optimization in road networks, SIAM J. Sci. Comput., Volume 39 (2017) no. 1, p. B53-B75 | DOI | MR | Zbl
[6] Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications, Industrial Mathematics and Complex Systems, Springer, 2017, pp. 77-122 | DOI | Zbl
[7] Time-optimal control of automobile test drives with gear shifts, Optim. Control Appl. Methods, Volume 31 (2010) no. 2, pp. 137-153 | DOI | MR | Zbl
[8] Sequential linear integer programming for integer optimal control with total variation regularization, ESAIM, Control Optim. Calc. Var., Volume 28 (2022), 66, 34 pages | MR | Zbl
Cited by Sources:
Comments - Policy