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Published: 1 June 2024

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1

Bongini, Mattia, Massimo Fornasier, Francesco Rossi, and Francesco Solombrino. "Mean-Field Pontryagin Maximum Principle." Journal of Optimization Theory and Applications 175, no. 1 (August 10, 2017): 1–38. http://dx.doi.org/10.1007/s10957-017-1149-5.

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2

Artstein, Zvi. "Pontryagin Maximum Principle Revisited with Feedbacks." European Journal of Control 17, no. 1 (January 2011): 46–54. http://dx.doi.org/10.3166/ejc.17.46-54.

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3

Avakov, E. R., and G. G. Magaril-Il’yaev. "Pontryagin maximum principle, relaxation, and controllability." Doklady Mathematics 93, no. 2 (March 2016): 193–96. http://dx.doi.org/10.1134/s1064562416020216.

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4

Cardin, Franco, and Andrea Spiro. "Pontryagin maximum principle and Stokes theorem." Journal of Geometry and Physics 142 (August 2019): 274–86. http://dx.doi.org/10.1016/j.geomphys.2019.04.014.

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5

Roth, Oliver. "Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions." Canadian Journal of Mathematics 67, no. 4 (August 1, 2015): 942–60. http://dx.doi.org/10.4153/cjm-2014-027-6.

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AbstractIn this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ℂn.

6

Lovison, Alberto, and Franco Cardin. "A Pareto–Pontryagin Maximum Principle for Optimal Control." Symmetry 14, no. 6 (June 6, 2022): 1169. http://dx.doi.org/10.3390/sym14061169.

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In this paper, an attempt to unify two important lines of thought in applied optimization is proposed. We wish to integrate the well-known (dynamic) theory of Pontryagin optimal control with the Pareto optimization (of the static type), involving the maximization/minimization of a non-trivial number of functions or functionals, Pontryagin optimal control offers the definitive theoretical device for the dynamic realization of the objectives to be optimized. The Pareto theory is undoubtedly less known in mathematical literature, even if it was studied in topological and variational details (Morse theory) by Stephen Smale. This reunification, obviously partial, presents new conceptual problems; therefore, a basic review is necessary and desirable. After this review, we define and unify the two theories. Finally, we propose a Pontryagin extension of a recent multiobjective optimization application to the evolution of trees and the related anatomy of the xylems . This work is intended as the first contribution to a series to be developed by the authors on this subject.

7

Agrachev, A. A., and R. V. Gamkrelidze. "The Pontryagin Maximum Principle 50 years later." Proceedings of the Steklov Institute of Mathematics 253, S1 (July 2006): S4—S12. http://dx.doi.org/10.1134/s0081543806050026.

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8

Grabowski, Janusz, and MichałJóźwikowski. "Pontryagin Maximum Principle on Almost Lie Algebroids." SIAM Journal on Control and Optimization 49, no. 3 (January 2011): 1306–57. http://dx.doi.org/10.1137/090760246.

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9

Ohsawa, Tomoki. "Contact geometry of the Pontryagin maximum principle." Automatica 55 (May 2015): 1–5. http://dx.doi.org/10.1016/j.automatica.2015.02.015.

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10

Magaril-Il’yaev, G. G. "The Pontryagin maximum principle: Statement and proof." Doklady Mathematics 85, no. 1 (February 2012): 14–17. http://dx.doi.org/10.1134/s1064562412010048.

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11

Dmitruk, A. V., and A. M. Kaganovich. "The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle." Systems & Control Letters 57, no. 11 (November 2008): 964–70. http://dx.doi.org/10.1016/j.sysconle.2008.05.006.

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12

Ndaïrou, Faïçal, and Delfim F. M. Torres. "Pontryagin Maximum Principle for Distributed-Order Fractional Systems." Mathematics 9, no. 16 (August 8, 2021): 1883. http://dx.doi.org/10.3390/math9161883.

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We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.

13

Otmane, Cherif Abdelillah. "APPLICATION OF THE PONTRYAGIN MAXIMUM PRINCIPLE TO CONTROL TUMOR ANGIOGENESIS." Chronos 6, no. 12(62) (December 13, 2021): 51–55. http://dx.doi.org/10.52013/2658-7556-62-12-16.

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We present a sample application covering several cases using an extension of the Pontryagin Minimum Principle (PMP) [3]. We are interested in the management of tumor angiogenesis, that is, the therapeutic management of the proliferation of cancer cells that develop new blood vessels. Let us formulate the problem and derive the optimal control and apply the Pontryagin maximum principle to our optimal trajectory, and we derive the theorem and check it with an example. Then we will study stabilization.

14

Otmane, Cherif Abdelillah. "APPLICATION OF THE PONTRYAGIN MAXIMUM PRINCIPLE TO CONTROL TUMOR ANGIOGENESIS." Chronos 7, no. 10(72) (November 13, 2022): 74–78. http://dx.doi.org/10.52013/2658-7556-72-10-22.

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We present a sample application covering several cases using an extension of the Pontryagin Minimum Principle (PMP) [3]. We are interested in the management of tumor angiogenesis, that is, the therapeutic management of the proliferation of cancer cells that develop new blood vessels. Let us formulate the problem and derive the optimal control and apply the Pontryagin maximum principle to our optimal trajectory, and we derive the theorem and check it with an example. Then we will study stabilization.

15

Ndaïrou, Faïçal, and Delfim F. M. Torres. "Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems." Mathematics 11, no. 19 (October 9, 2023): 4218. http://dx.doi.org/10.3390/math11194218.

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We introduce a new optimal control problem where the controlled dynamical system depends on multi-order (incommensurate) fractional differential equations. The cost functional to be maximized is of Bolza type and depends on incommensurate Caputo fractional-orders derivatives. We establish continuity and differentiability of the state solutions with respect to perturbed trajectories. Then, we state and prove a Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems. Finally, we give an example, illustrating the applicability of our Pontryagin maximum principle.

16

Krastanov, M. I., N. K. Ribarska, and Ts Y. Tsachev. "A Pontryagin Maximum Principle for Infinite-Dimensional Problems." SIAM Journal on Control and Optimization 49, no. 5 (January 2011): 2155–82. http://dx.doi.org/10.1137/100799009.

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17

Avakov, E. R., and G. G. Magaril-Il’yaev. "Mix of Controls and the Pontryagin Maximum Principle." Journal of Mathematical Sciences 217, no. 6 (August 13, 2016): 672–82. http://dx.doi.org/10.1007/s10958-016-2999-3.

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18

Ioffe, Alexander D. "An Elementary Proof of the Pontryagin Maximum Principle." Vietnam Journal of Mathematics 48, no. 3 (March 5, 2020): 527–36. http://dx.doi.org/10.1007/s10013-020-00397-0.

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19

Gomoyunov, M. I. "On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems." Дифференциальные уравнения 59, no. 11 (December 15, 2023): 1515–21. http://dx.doi.org/10.31857/s0374064123110067.

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We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.

20

Nainggolan, J., F. J. Iswar, and Abraham Abraham. "KONTROL OPTIMAL PADA PEYEBARAN TUBERKULOSIS DENGAN EXOGENOUS REINFECTION." JURNAL ILMIAH MATEMATIKA DAN TERAPAN 16, no. 1 (May 20, 2019): 42–50. http://dx.doi.org/10.22487/2540766x.2019.v16.i1.12762.

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Tuberculosis is a disease caused by Mycobacterium tuberculosis. Tuberculosis can be controlled through treatment, chemoprophylaxis and vaccination. Optimal control of treatment in the exposed compartment can be done in an effort to reduce the number of exposed compartments individual into the active compartment of tuberculosis. Optimal control can be completed by the Pontryagin Maximum Principle Method. Based on numerical simulation results, optimal control of treatment in the exposed compartment can reduce the number of infected compartments individual with active TB.Keywords : Exogenous Reinfection, Optimal Control, Pontryagin's Maximum Principle, Spread Of Tuberculosis.

21

Rodrigues, Hugo Murilo, and Ryuichi Fukuoka. "Geodesic fields for Pontryagin type C⁰-Finsler manifolds." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 19. http://dx.doi.org/10.1051/cocv/2022013.

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Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.

22

Fakharzadeh, Alireza, Somayeh Sharif, and Karim Eslamloueyan. "Extension Of Pontryagin Maximum Principle And Its Economical Applications." Journal of Mathematics and Computer Science 05, no. 04 (December 30, 2012): 313–19. http://dx.doi.org/10.22436/jmcs.05.04.09.

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23

Dehaghani, Nahid Binandeh, and Fernando Lobo Pereira. "High Fidelity Quantum State Transfer by Pontryagin Maximum Principle." IFAC-PapersOnLine 55, no. 16 (2022): 214–19. http://dx.doi.org/10.1016/j.ifacol.2022.09.026.

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24

Liu, Xuan. "A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket." SIAM Journal on Control and Optimization 56, no. 6 (January 2018): 4288–308. http://dx.doi.org/10.1137/17m1113606.

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25

Aseev, S. M., and A. V. Kryazhimskii. "The Pontryagin maximum principle and optimal economic growth problems." Proceedings of the Steklov Institute of Mathematics 257, no. 1 (July 2007): 1–255. http://dx.doi.org/10.1134/s0081543807020010.

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26

Magaril-Il’yaev, G. G. "The Pontryagin maximum principle. Ab ovo usque ad mala." Proceedings of the Steklov Institute of Mathematics 291, no. 1 (November 2015): 203–18. http://dx.doi.org/10.1134/s0081543815080167.

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27

Gamkrelidze, R. V. "History of the Discovery of the Pontryagin Maximum Principle." Proceedings of the Steklov Institute of Mathematics 304, no. 1 (January 2019): 1–7. http://dx.doi.org/10.1134/s0081543819010012.

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28

Quoc Khanh, Phan. "On Pontryagin maximum principle for linear systems with constraints." Optimization 16, no. 1 (January 1985): 63–69. http://dx.doi.org/10.1080/02331938508842990.

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29

Little, G. "The Pontryagin maximum principle: the constancy of the Hamiltonian." IMA Journal of Mathematical Control and Information 13, no. 4 (December 1, 1996): 403–8. http://dx.doi.org/10.1093/imamci/13.4.403.

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30

Bergounioux, Maïtine, and Housnaa Zidani. "Pontryagin Maximum Principle for Optimal Control of Variational Inequalities." SIAM Journal on Control and Optimization 37, no. 4 (January 1999): 1273–90. http://dx.doi.org/10.1137/s0363012997328087.

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31

Ballestra, Luca Vincenzo. "The spatial AK model and the Pontryagin maximum principle." Journal of Mathematical Economics 67 (December 2016): 87–94. http://dx.doi.org/10.1016/j.jmateco.2016.09.012.

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32

Waldherr, Steffen, and Henning Lindhorst. "Optimality in cellular storage via the Pontryagin Maximum Principle." IFAC-PapersOnLine 50, no. 1 (July 2017): 9889–95. http://dx.doi.org/10.1016/j.ifacol.2017.08.1615.

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33

Lin, Ping, and Jiongmin Yong. "Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle." SIAM Journal on Control and Optimization 58, no. 1 (January 2020): 136–64. http://dx.doi.org/10.1137/19m124602x.

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34

Kamocki, Rafal. "Pontryagin maximum principle for fractional ordinary optimal control problems." Mathematical Methods in the Applied Sciences 37, no. 11 (July 10, 2013): 1668–86. http://dx.doi.org/10.1002/mma.2928.

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35

Bourdin, Loïc, and Emmanuel Trélat. "Pontryagin maximum principle for optimal sampled-data control problems." IFAC-PapersOnLine 48, no. 25 (2015): 80–84. http://dx.doi.org/10.1016/j.ifacol.2015.11.063.

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36

Zhou, Jiangjing, Anna Tur, Ovanes Petrosian, and Hongwei Gao. "Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle." Mathematics 9, no. 2 (January 14, 2021): 163. http://dx.doi.org/10.3390/math9020163.

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We consider a class of cooperative differential games with continuous updating making use of the Pontryagin maximum principle. It is assumed that at each moment, players have or use information about the game structure defined in a closed time interval of a fixed duration. Over time, information about the game structure will be updated. The subject of the current paper is to construct players’ cooperative strategies, their cooperative trajectory, the characteristic function, and the cooperative solution for this class of differential games with continuous updating, particularly by using Pontryagin’s maximum principle as the optimality conditions. In order to demonstrate this method’s novelty, we propose to compare cooperative strategies, trajectories, characteristic functions, and corresponding Shapley values for a classic (initial) differential game and a differential game with continuous updating. Our approach provides a means of more profound modeling of conflict controlled processes. In a particular example, we demonstrate that players’ behavior is braver at the beginning of the game with continuous updating because they lack the information for the whole game, and they are “intrinsically time-inconsistent”. In contrast, in the initial model, the players are more cautious, which implies they dare not emit too much pollution at first.

37

Mamedova, Turkan. "ANALOGUE OF THE DISCRETE MAXIMUM PRINCIPLE AND THE NECESSARY OPTIMALITY CONDITION OF SINGULAR CONTROLS IN ONE TWO-PARAMETRIC DISCRETE OPTIMAL CONTROL PROBLEM." Applied Mathematics and Control Sciences, no. 3 (November 10, 2021): 7–34. http://dx.doi.org/10.15593/2499-9873/2021.3.01.

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A two-stage (stepwise) optimal control problem for linear two-parameter systems with distributed control functions is considered. The aim of the work is to establish the necessary optimality condition under the assumption that the convexity of the set of admissible controls is satisfied and the connection condition is nonlinear. Using increments of the quality functional in the form of two-dimensional linear inhomogeneous systems of difference equations, a formula is obtained that allows one to obtain both a discrete analogue of the Pontryagin maximum principle and to study the case of its degeneration. A theorem is formulated that is an analogue of the discrete Pontryagin maximum principle for the problem under consideration. In the case of special controls, the discrete maximum principle degenerates and, therefore, becomes ineffective, including in the verification sense. Therefore, it is necessary to have new necessary conditions for optimality. A special, in the sense of the Pontryagin maximum principle, case of a discrete maximum condition, under which admissible controls are considered special, is studied. A necessary condition for optimality of singular controls is established.

38

Rasulzade, Shahla. "REQUIRED OPTIMALITY CONDITIONS IN ONE OPTIMAL CONTROL PROBLEM WITH MULTIPOINT FUNCTIONAL." Applied Mathematics and Control Sciences, no. 2 (June 30, 2020): 7–26. http://dx.doi.org/10.15593/2499-9873/2020.2.01.

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One specific optimal control problem with distributed parameters of the Moskalenko type with a multipoint quality functional is considered. To date, the theory of necessary first-order optimality conditions such as the Pontryagin maximum principle or its consequences has been sufficiently developed for various optimal control problems described by ordinary differential equations, i.e. for optimal control problems with lumped parameters. Many controlled processes are described by various partial differential equations (processes with distributed parameters). Some features are inherent in optimal control problems with distributed parameters, and therefore, when studying the optimal control problem with distributed parameters, in particular, when deriving various necessary optimality conditions, non-trivial difficulties arise. In particular, in the study of cases of degeneracy of the established necessary optimality conditions, fundamental difficulties arise. In the present work, we study one optimal control problem described by a system of first-order partial differential equations with a controlled initial condition under the assumption that the initial function is a solution to the Cauchy problem for ordinary differential equations. The objective function (quality criterion) is multi-point. Therefore, it becomes necessary to introduce an unconventional conjugate equation, not in differential (classical), but in integral form. In the work, using one version of the increment method, using the explicit linearization method of the original system, the necessary optimality condition is proved in the form of an analog of the maximum principle of L.S. Pontryagin. It is known that the maximum principle of L.S. Pontryagin for various optimal control problems is the strongest necessary condition for optimality. But the principle of a maximum of L.S. Pontryagin, being a necessary condition of the first order, often degenerates. Such cases are called special, and the corresponding management, special management. Based on these considerations, in the considered problem, we study the case of degeneration of the maximum principle of L.S. Pontryagin for the problem under consideration. For this purpose, a formula for incrementing the quality functional of the second order is constructed. By introducing auxiliary matrix functions, it was possible to obtain a second-order increment formula that is constructive in nature. The necessary optimality condition for special controls in the sense of the maximum principle of L.S. Pontryagin is proved. The proved necessary optimality conditions are explicit.

39

Zemliak, Alexander. "Analysis of strategies of circuit optimisation on basis of maximum principle." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 1 (January 2, 2018): 484–503. http://dx.doi.org/10.1108/compel-12-2016-0540.

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Purpose This paper aims to propose a new approach on the problem of circuit optimisation by using the generalised optimisation methodology presented earlier. This approach is focused on the application of the maximum principle of Pontryagin for searching the best structure of a control vector providing the minimum central processing unit (CPU) time. Design/methodology/approach The process of circuit optimisation is defined mathematically as a controllable dynamical system with a control vector that changes the internal structure of the equations of the optimisation procedure. In this case, a well-known maximum principle of Pontryagin is the best theoretical approach for finding of the optimum structure of control vector. A practical approach for the realisation of the maximum principle is based on the analysis of the behaviour of a Hamiltonian for various strategies of optimisation and provides the possibility to find the optimum points of switching for the control vector. Findings It is shown that in spite of the fact that the maximum principle is not a sufficient condition for obtaining the global minimum for the non-linear problem, the decision can be obtained in the form of local minima. These local minima provide rather a low value of the CPU time. Numerical results were obtained for both a two-dimensional case and an N-dimensional case. Originality/value The possibility of the use of the maximum principle of Pontryagin to a problem of circuit optimisation is analysed systematically for the first time. The important result is the theoretical justification of formerly discovered effect of acceleration of the process of circuit optimisation.

40

Aliyeva, Saadat Tofig. "The Pontryagin maximum principle for nonlinear fractional order difference equations." Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitel'naya tekhnika i informatika, no. 54 (March 1, 2021): 4–11. http://dx.doi.org/10.17223/19988605/54/1.

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41

Golvinskii, P. A. "Pontryagin principle of maximum for the quantum problem of speed." Automation and Remote Control 68, no. 4 (April 2007): 610–18. http://dx.doi.org/10.1134/s0005117907040042.

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42

Gamkrelidze, R. V. "Dual formulation of the Pontryagin maximum principle in optimal control." Proceedings of the Steklov Institute of Mathematics 291, no. 1 (November 2015): 61–67. http://dx.doi.org/10.1134/s0081543815080064.

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43

Paruchuri, Pradyumna, and Debasish Chatterjee. "Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints." IEEE Transactions on Automatic Control 64, no. 10 (October 2019): 4202–8. http://dx.doi.org/10.1109/tac.2019.2893160.

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44

Phogat, Karmvir Singh, Debasish Chatterjee, and Ravi N. Banavar. "A discrete-time Pontryagin maximum principle on matrix Lie groups." Automatica 97 (November 2018): 376–91. http://dx.doi.org/10.1016/j.automatica.2018.08.026.

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45

Barron, E. N. "The Pontryagin maximum principle for minimax problems of optimal control." Nonlinear Analysis: Theory, Methods & Applications 15, no. 12 (December 1990): 1155–65. http://dx.doi.org/10.1016/0362-546x(90)90051-h.

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46

Blot, Joël, and Hasan Yilmaz. "A Generalization of Michel’s Result on the Pontryagin Maximum Principle." Journal of Optimization Theory and Applications 183, no. 3 (September 26, 2019): 792–812. http://dx.doi.org/10.1007/s10957-019-01587-8.

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47

Kipka, Robert J., and Yuri S. Ledyaev. "Pontryagin Maximum Principle for Control Systems on Infinite Dimensional Manifolds." Set-Valued and Variational Analysis 23, no. 1 (October 4, 2014): 133–47. http://dx.doi.org/10.1007/s11228-014-0301-8.

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48

Chang, Dong Eui. "A simple proof of the Pontryagin maximum principle on manifolds." Automatica 47, no. 3 (March 2011): 630–33. http://dx.doi.org/10.1016/j.automatica.2011.01.037.

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49

Sumin, Mikhail Iosifovich. "WHY REGULARIZATION OF LAGRANGE PRINCIPLE AND PONTRYAGIN MAXIMUM PRINCIPLE IS NEEDED AND WHAT IT GIVES." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 757–75. http://dx.doi.org/10.20310/1810-0198-2018-23-124-757-775.

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We consider the regularization of the classical Lagrange principle and the Pontryagin maximum principle in convex problems of mathematical programming and optimal control. On example of the “simplest” problems of constrained infinitedimensional optimization, two main questions are discussed: why is regularization of the classical optimality conditions necessary and what does it give?

50

Rivera, Javier. "A Note on a Possible Connection between Pontryagin Maximum Principle and Jaynes Maximum Entropy." Annals of Pure and Applied Mathematics 17, no. 2 (May 1, 2018): 233–40. http://dx.doi.org/10.22457/apam.v17n2a9.

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