{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:36:08Z","timestamp":1760146568887,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T00:00:00Z","timestamp":1731628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schr\u00f6dinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1\/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1\/2 particle system.<\/jats:p>","DOI":"10.3390\/e26110978","type":"journal-article","created":{"date-parts":[[2024,11,15]],"date-time":"2024-11-15T05:33:04Z","timestamp":1731648784000},"page":"978","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum Control Design by Lyapunov Trajectory Tracking and Optimal Control"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8259-5241","authenticated-orcid":false,"given":"Hongli","family":"Yang","sequence":"first","affiliation":[{"name":"College of Big Data, Qingdao Huanghai University, Qingdao 266427, China"},{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Guohui","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9019-072X","authenticated-orcid":false,"given":"Ivan Ganchev","family":"Ivanov","sequence":"additional","affiliation":[{"name":"Faculty of Economics and Business Administration, St. Kl. Ohridski Sofia University, 1113 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1119\/1.5034785","article-title":"Blackbody radiation in classical physics: A historical perspective","volume":"86","author":"Timothy","year":"2018","journal-title":"Am. J. Phys."},{"key":"ref_2","first-page":"132","article-title":"Adiabatic invariance and its application to Wien\u2019s complete displacement law of blackbody radiation","volume":"86","author":"Don","year":"2024","journal-title":"Am. J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"710","DOI":"10.1119\/5.0137631","article-title":"A demonstration of the photoelectric effect with sunlight","volume":"61","author":"Gkourmpis","year":"2023","journal-title":"Phys. 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