{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:45:03Z","timestamp":1759970703428,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T00:00:00Z","timestamp":1738022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Regional Development Fund under the Operational Program \u201cScientific Research, Innovation and Digitization for Smart Transformation 2021\u20132027\u201d, Project CoC \u201cSmart Mechatronics, Eco- and Energy Saving Systems and Technologies\u201d","award":["BG16RFPR002-1.014-0005"],"award-info":[{"award-number":["BG16RFPR002-1.014-0005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This purely theoretical study considers two new geometric state-space properties of the classical linear time-optimal control problem with one input and real non-positive eigenvalues of the system, with constraints only on the control input and without constraints on the state-space variables, following from Pontryagin\u2019s maximum principle. These properties complement the well-known facts from the maximum principle about the number of switchings of the control function and the character of the optimal phase trajectories of the system leading it to the state-space origin. They lay the foundation of a new method for synthesizing the time-optimal control without the need to describe the switching hyper-surfaces. The new technique is demonstrated on two examples. The so-called \u201caxes initialization\u201d and the synthesis technique are illustrated on the double integrator system in its entirety. The second one is on a hypothetical seventh-order system.<\/jats:p>","DOI":"10.3390\/axioms14020097","type":"journal-article","created":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T08:57:48Z","timestamp":1738054668000},"page":"97","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some New Geometric State-Space Properties of the Classical Linear Time-Optimal Control Problem with One Input and Real Non-Positive Eigenvalues of the System Following from Pontryagin\u2019s Maximum Principle"],"prefix":"10.3390","volume":"14","author":[{"given":"Borislav G.","family":"Penev","sequence":"first","affiliation":[{"name":"Department of Control Systems, Faculty of Electronics and Automatics, Plovdiv Branch, Technical University of Sofia, 4000 Plovdiv, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,28]]},"reference":[{"key":"ref_1","unstructured":"Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mischenko, E.F. (1964). The Mathematical Theory of Optimal Processes, Pergamon Press."},{"key":"ref_2","first-page":"5","article-title":"The simplest relay automatic control systems","volume":"X","author":"Feldbaum","year":"1949","journal-title":"Autom. Remote Control."},{"key":"ref_3","first-page":"6","article-title":"Optimal processes in automatic control systems","volume":"XIV","author":"Feldbaum","year":"1953","journal-title":"Autom. Remote Control."},{"key":"ref_4","unstructured":"Athans, M., and Falb, P.L. (1966). Optimal control. An Introduction to the Theory and Its Applications, McGraw-Hill."},{"key":"ref_5","unstructured":"Lee, E.B., and Markus, L. (1967). Foundations of Optimal Control Theory, Wiley & Sons Inc."},{"key":"ref_6","unstructured":"Bryson, A.E., and Ho, Y.C. (1969). Applied Optimal Control, Blaisdell Publishing Company."},{"key":"ref_7","unstructured":"\u0411o\u043b\u0442\u044f\u043d\u0441\u043a\u0438\u0439, \u0412.\u0413. (1969). \u041c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u0447\u0435\u0441\u043a\u0438\u0435 \u041c\u0435\u0442o\u0434\u044b \u041e\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043do\u0433o \u0423\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u044f, \u041d\u0430\u0443\u043a\u0430."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Leitmann, G. (1981). The Calculus of Variations and Optimal Control, Plenum Press.","DOI":"10.1007\/978-1-4899-0333-4"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Pinch, E.R. (1993). Optimal Control and the Calculus of Variations, Oxford University Press.","DOI":"10.1093\/oso\/9780198532170.001.0001"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Locatelli, A. (2017). Optimal Control of a Double Integrator, Springer. Studies in Systems, Decision and Control.","DOI":"10.1007\/978-3-319-42126-1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109151","DOI":"10.1016\/j.automatica.2020.109151","article-title":"Time-optimal control of linear time invariant systems between two arbitrary states","volume":"120","author":"Romano","year":"2020","journal-title":"Automatica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109240","DOI":"10.1016\/j.automatica.2020.109240","article-title":"Time optimal control of triple integrator with input saturation and full state constraints","volume":"122","author":"He","year":"2020","journal-title":"Automatica"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Consolini, L., and Piazzi, A. (2006, January 13\u201315). Generalized bang-bang control for feedforward constrained regulation. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA.","DOI":"10.1109\/CDC.2006.377142"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2234","DOI":"10.1016\/j.automatica.2009.06.030","article-title":"Generalized bang-bang control for feedforward constrained regulation","volume":"45","author":"Consolini","year":"2009","journal-title":"Automatica"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Consolini, L., Laurini, M., and Piazzi, A. (2024, January 11\u201314). Generalized Bang-Bang Control for Multivariable Feedforward Regulation. Proceedings of the 32nd Mediterranean Conference on Control and Automation (MED), Chania, Greece.","DOI":"10.1109\/MED61351.2024.10566231"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3878","DOI":"10.1109\/LCSYS.2023.3344522","article-title":"Efficient Near-Optimal Control of Large-Size Second-Order Linear Time-Varying Systems","volume":"7","author":"Rustagi","year":"2023","journal-title":"IEEE Control. Syst. Lett."},{"key":"ref_17","first-page":"8","article-title":"Time-Optimal Planning for Long-Range Quadrotor Flights: An Automatic Optimal Synthesis Approach","volume":"14","author":"Qin","year":"2021","journal-title":"J. Latex Cl. Files"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1137\/0312046","article-title":"The Closed-Loop Time-Optimal Control. I: Optimality","volume":"12","year":"1974","journal-title":"SIAM J. Control."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1137\/0325062","article-title":"Regular Synthesis for Time-Optimal Control of Single-Input Real Analytic Systems in the Plane","volume":"25","author":"Sussmann","year":"1987","journal-title":"SIAM J. Control. Optim."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Penev, B. (2023). One New Property of a Class of Linear Time-Optimal Control Problems. Mathematics, 11.","DOI":"10.20944\/preprints202306.0840.v1"},{"key":"ref_21","unstructured":"(2024, October 15). L7.3 Time-Optimal Control for Linear Systems Using Pontryagin\u2019s Principle of Maximum, Graduate Course on \u201cOptimal and Robust Control\u201d (B3M35ORR, BE3M35ORR, BEM35ORC) Given at Faculty of Electrical Engineering, Czech Technical University in Prague. Available online: https:\/\/www.youtube.com\/watch?v=YiIksQcg8EU."},{"key":"ref_22","unstructured":"(2024, October 15). Solution of Minimum\u2014Time Control Problem with an Example. Available online: https:\/\/www.youtube.com\/watch?v=Oi90M3cS8wg."},{"key":"ref_23","unstructured":"(2024, October 15). Alexandre Girard, Commande Optimale: Exemple Pour le Temps Minimum d\u2019une Masse avec une Limite de Force, Mod\u00e9lisation, Analyse et Commande des Robots, Exemple d\u2019un Calcul de loi de Commande Optimale. Available online: https:\/\/www.youtube.com\/watch?v=wKjEAXFvXlQ."},{"key":"ref_24","unstructured":"\u0420\u044e\u0442\u0438\u043d, \u041a.C. (2024, October 15). \u0412\u0430\u0440\u0438\u0430\u0446\u0438o\u043d\u043do\u0435 \u0438\u0441\u0447\u0438\u0441\u043b\u0435\u043d\u0438\u0435 \u0438 o\u043f\u0442\u0438\u043c\u0430\u043b\u044c\u043do\u0435 \u0443\u043f\u0440\u0430\u0432\u043b\u0435\u043d\u0438\u0435\u201412. \u0417\u0430\u0434\u0430\u0447\u0430 \u0431\u044b\u0441\u0442\u0440o\u0434\u0435\u0439\u0441\u0442\u0432\u0438\u044f. Available online: https:\/\/www.youtube.com\/watch?v=u7FtLP5BWeg."},{"key":"ref_25","unstructured":"Penev, B.G. (2012). Analysis of the possibility for time-optimal control of the scanning system of the GREEN-WAKE\u2019s project lidar. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Glattfelder, A.H., and Schaufelberger, W. (2003). Control systems with input and output constraints, Springer.","DOI":"10.1007\/978-1-4471-0047-8"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.neucom.2015.12.078","article-title":"Nonlinear time-delay anti-windup compensator synthesis for nonlinear time-delay systems: A delay-range-dependent approach","volume":"186","author":"Hussain","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1007\/s11071-018-4666-3","article-title":"Static anti-windup compensator design for nonlinear time-delay systems subjected to input saturation","volume":"95","author":"Hussain","year":"2019","journal-title":"Nonlinear Dyn."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/97\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:37:32Z","timestamp":1759919852000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/97"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,28]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020097"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020097","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,28]]}}}