Abstract
The paper is devoted to the study of a new class of optimal control problems governed by discontinuous constrained differential inclusions of the sweeping type involving the duration of the dynamic process into optimization. We develop a novel version of the method of discrete approximations of its own qualitative and numerical values with establishing its well-posedness and strong convergence to optimal solutions of the controlled sweeping process. Using advanced tools of first-order and second-order variational analysis and generalized differentiation allows us to derive new necessary conditions for optimal solutions of the discrete-time problems and then, by passing to the limit in the discretization procedure, for designated local minimizers in the original problem of sweeping optimal control. The obtained results are illustrated by a numerical example.
Similar content being viewed by others
Notes
Here we mean the union of the density points of E such that the normal cone to C(t) is active along E.
References
Adam, L., Outrata, J.V.: On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete Contin. Dyn. Syst. Ser. B 19, 2709–2738 (2014)
Adly, S., Haddad, T., Thibault, L.: Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities. Math. Program. 148, 5–47 (2014)
Arroud, C.E., Colombo, G.: A maximum principle of the controlled sweeping process. Set-Valued Var. Anal. 26, 607–629 (2018)
Aubin, J.P.: Viability Theory. Birkhäuser, Boston (1991)
Bounkhel, M.: Mathematical modeling and numerical simulations of the motion of nanoparticles in straight tube. Adv. Mech. Eng. 8, 1–6 (2016)
Brogliato, B., Tanwani, A.: Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability. SIAM Rev. 62, 3–129 (2020)
Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discrete Contin. Dyn. Syst. Ser. B 18, 331–348 (2013)
Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)
Cao, T.H., Mordukhovich, B.S.: Optimal control of a nonconvex perturbed sweeping process. J. Diff. Eqs. 266, 1003–1050 (2019)
Cao, T.H., Mordukhovich, B.S.: Applications of optimal control of a nonconvex sweeping processes to optimization of the planar crowd motion model. Discrete Contin. Dyn. Syst. Ser. B 24, 4191–4216 (2019)
Cao, T.H., Colombo, G., Mordukhovich, B.S., Nguyen, D.: Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations. J. Diff. Eqs. 274, 461–509 (2021)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Colombo, G., Gidoni, P.: On the optimal control of rate-independent soft crawlers. J. Math. Pures Appl. 146, 127–157 (2021)
Colombo, G., Henrion, R., Hoang, N.D., Mordukhovich, B.S.: Optimal control of the sweeping process over polyhedral controlled sets. J. Diff. Eqs. 260, 3397–3447 (2016)
Colombo, G., Mordukhovich, B.S., Nguyen, D.: Optimal control of sweeping processes in robotics and traffic flow models. J. Optim. Theory Appl. 182, 439–472 (2019)
Colombo, G., Mordukhovich, B.S., Nguyen, D.: Optimization of a perturbed sweeping process by discontinuous controls. SIAM J. Control Optim. 58, 2678–2709 (2020)
de Pinho, M.D.R., Ferreira, M.M.A., Smirnov, G.V.: Optimal control involving sweeping processes. Set-Valued Var. Anal. 27, 523–548 (2019)
de Pinho, M.D.R., Ferreira, M.M.A., Smirnov, G.V.: A maximum principle for optimal control problems involving sweeping processes with a nonsmooth set. J. Optim. Theory Appl. 199, 273–297 (2023)
Hedjar, R., Bounkhel, M.: An automatic collision avoidance algorithm for multiple marine surface vehicles. Int. J. Appl. Math. Comput. Sci. 29, 759–768 (2019)
Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)
Henrion, R., Jourani, A., Mordukhovich, B.S.: Controlled polyhedral sweeping processes: existence, stability, and optimality conditions. J. Diff. Eqs. 336, 408–443 (2023)
Hermosilla, C., Palladino, M.: Optimal control of the sweeping process with a nonsmooth moving set. SIAM J. Control 60, 2811–2834 (2022)
Khalil, N.T., Pereira, F.L.: A maximum principle for state-constrained optimal sweeping control problems. IEEE Contr. Syst. Lett. 7, 43–48 (2022)
Krasnosel’skiǐ, M.A., Pokrovskiǐ, A.V.: Systems with Hysteresis. Springer, New York (1989)
Monteiro Marques, M.D.P.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Birkhäuser, Basel (1993)
Mordukhovich, B.S., Nguyen D., Nguyen, T.: Optimal control of sweeping processes in unmanned surface vehicles and nanoparticle modeling; arXiv:2311.12916 (2023)
Mordukhovich, B.S.: Optimization and finite difference approximations of nonconvex differential inclusions with free time. In: Mordukhovich, B.S., Sussmann, H.J. (eds.) Nonsmooth Analysis and Geometric Methods in Deterministic Optimal Control, pp. 153–202. Springer, New York
Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, vol. 58, pp. 32–46. Philadelphia, PA (1992)
Mordukhovich, B.S.: Discrete approximations and refined Euler-Lagrange conditions for differential inclusions. SIAM J. Control Optim. 33, 882–915 (1995)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational Analysis and Applications. Springer, Cham, Switzerland (2018)
Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988, 2nd edn. URSS Publishing, Moscow (2023)
Mordukhovich, B.S.: Second-Order Variational Analysis in Optimization, Variational Stability and Control: Theory, Algorithms, Applications. Springer, Cham, Switzerland (2024)
Mordukhovich, B.S., Nguyen, D.: Discrete approximations and optimal control of nonsmooth perturbed sweeping processes. J. Convex Anal. 28, 655–688 (2021)
Moreau, J.J.: On unilateral constraints, friction and plasticity. In: Capriz, G., Stampacchia, G. (eds.) New Variational Techniques in Mathematical Physics, pp. 173–322. Proc. C.I.M.E. Summer Schools, Cremonese, Rome (1974)
Nour, C., Zeidan, V.: Pontryagin-type maximum principle for a controlled sweeping process with nonsmooth and unbounded sweeping set, J. Convex Anal., to appear (2024)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Stewart, D.E.: Dynamics with Inequalities: Impacts and Hard Constraints. SIAM, Philadelphia, PA (2011)
Vinter, R.B.: Optimal Control. Birkhaüser, Boston (2000)
Zeidan, V., Nour, C., Saoud, H.: A nonsmooth maximum principle for a controlled nonconvex sweeping process. J. Diff. Eqs. 269, 9531–9582 (2020)
Funding
The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors declare that the presented results are new, and there is no any conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of G. Colombo was partly supported by the project funded by the EuropeanUnion - NextGenerationEU under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.1 - Call PRIN 2022 No. 104 of February 2, 2022 of Italian Ministry of University and Research; Project 2022238YY5 (subject area: PE - Physical Sciences and Engineering) “Optimal control problems: analysis, approximation and applications”.
Research of B. S. Mordukhovich was partially supported by the USA National Science Foundation under grants DMS-1808978 and DMS-2204519, by the Australian Research Council under grant DP-190100555, and by Project 111 of China under grant D21024.
Research of D. Nguyen was supported by the AMS-Simon Foundation.
Research of T. Nguyen was partially supported by the USA National Science Foundation under grant DMS-1808978 and DMS-2204519.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Colombo, G., Mordukhovich, B.S., Nguyen, D. et al. Discrete Approximations and Optimality Conditions for Controlled Free-Time Sweeping Processes. Appl Math Optim 89, 40 (2024). https://doi.org/10.1007/s00245-024-10108-7
Accepted:
Published:
DOI: https://doi.org/10.1007/s00245-024-10108-7
Keywords
- Optimal control
- Sweeping processes
- Variational analysis
- Generalized differentiation
- Discrete approximations
- Necessary optimality conditions