### AIMS Mathematics

2024, Issue 3: 6109-6144. doi: 10.3934/math.2024299
Research article Special Issues

# The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

• Received: 02 November 2023 Revised: 19 January 2024 Accepted: 29 January 2024 Published: 02 February 2024
• MSC : 26A33, 47J35, 58E30

• In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $\mathsf{X}$-valued left Caputo fractional evolution equation with the analytic semigroup, where $\mathsf{X}$ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $\mathsf{X}$. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $\mathsf{X}^*$, the dual space of $\mathsf{X}$. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.

Citation: Yuna Oh, Jun Moon. The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints[J]. AIMS Mathematics, 2024, 9(3): 6109-6144. doi: 10.3934/math.2024299

### Related Papers:

• In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $\mathsf{X}$-valued left Caputo fractional evolution equation with the analytic semigroup, where $\mathsf{X}$ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $\mathsf{X}$. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $\mathsf{X}^*$, the dual space of $\mathsf{X}$. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.

 [1] H. M. Ali, F. L. Pereira, S. M. A. Gamma, A new approach to the Pontryagin maximum principle for nonlinear fractional optimal control problems, Math. Methods Appl. Sci., 39 (2016), 3640–3649. https://doi.org/10.1002/mma.3811 doi: 10.1002/mma.3811 [2] R. Almeida, R. Kamocki, A. B. Malinowska, T. Odzijewicz, On the necessary optimality conditions for the fractional Cucker-Smale optimal control problem, Commun. Nonlinear Sci. Numer. Simul., 96 (2021), 105678. https://doi.org/10.1016/j.cnsns.2020.105678 doi: 10.1016/j.cnsns.2020.105678 [3] E. G. Bajlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Technische Universiteit Eindhoven, 2001. [4] M. Bergounioux, L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM Control Optim. Calc. Var., 26 (2020), 35. https://doi.org/10.1051/cocv/2019021 doi: 10.1051/cocv/2019021 [5] M. Bergounioux, H. Zidani, Pontryagin maximum principle for optimal control of variational inequalities, SIAM J. Control Optim., 37 (1999), 1273–1290. https://doi.org/10.1137/S0363012997328087 doi: 10.1137/S0363012997328087 [6] V. I. Bogachev, Measure theory, Springer, 2000. https://doi.org/10.1007/978-3-540-34514-5 [7] L. Bourdin, Cauchy-Lipschitz theory for fractional multi-order dynamics-state-transition matrices, Duhamel formulas and duality theorems, Differ. Integral Equ., 31 (2017), 559–594. https://doi.org/10.57262/die/1526004031 doi: 10.57262/die/1526004031 [8] R. Chaudhary, S. Reich, Existence and controllability results for Hilfer fractional evolution equations via integral contractors, Fract. Calc. Appl. Anal., 25 (2022), 2400–2419. https://doi.org/10.1007/s13540-022-00099-z doi: 10.1007/s13540-022-00099-z [9] P. Chen, Y. Li, Q. Chen, B. Feng, On the initial value problem of fractional evolution equations with noncompact semigroup, Comput. Math. Appl., 67 (2014), 1108–1115. https://doi.org/10.1016/j.camwa.2014.01.002 doi: 10.1016/j.camwa.2014.01.002 [10] F. H. Clarke, Optimization and nonsmooth analysis, SIAM, 1990. https://doi.org/10.1137/1.9781611971309 [11] K. Diethelm, The analysis of fractional differential equations, An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2 [12] X. L. Ding, I. Area, J. Nieto, Controlled singular evolution equations and Pontryagin type maximum principle with applications, Evol. Equ. Control The., 11 (2022), 1655–1679. https://doi.org/10.3934/eect.2021059 doi: 10.3934/eect.2021059 [13] G. Fabbri, F. Gozzi, A. Świȩch, Stochastic optimal control in infinite dimension, Dynamic Programming and HJB Equations, Springer, 2017. https://doi.org/10.1007/978-3-319-53067-3 [14] H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems, Appl. Math. Optim., 15 (1987), 141–185. https://doi.org/10.1007/BF01442651 doi: 10.1007/BF01442651 [15] H. O. Fattorini, Infinite dimensional optimization and control theory, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9780511574795 [16] T. M. Flett, Differential analysis, Cambridge University Press, 1980. https://doi.org/10.1017/CBO9780511897191 [17] H. Frankowska, E. M. Marchini, M. Mazzola, Necessary optimality conditions for infinite dimensional state constrained control problems, J. Differ. Equations, 264 (2018), 7294–7327. https://doi.org/10.1016/j.jde.2018.02.012 doi: 10.1016/j.jde.2018.02.012 [18] C. G. Gal, M. Warma, Fractional-in-time semilinear parabolic equations and applications, Springer, 2020. https://doi.org/10.1007/978-3-030-45043-4 [19] M. Gomoyunov, On representation formulas for solutions of linear differential equations with Caputo fractional derivatives, Fract. Calc. Appl. Anal., 23 (2020), 1141–1160. https://doi.org/10.1515/fca-2020-0058 doi: 10.1515/fca-2020-0058 [20] H. Hassani, A. Avazzadeh, P. Agarwal, M. Javad Ebadi, A. Bayati Eshkaftaki, Generalized Bernoulli-Laguerre polynomials: applications in coupled nonlinear system of variable-order fractional PDEs, J. Optim. Theory Appl., 200 (2023), 371–393. https://doi.org/10.1007/s10957-023-02346-6 doi: 10.1007/s10957-023-02346-6 [21] B. Hu, J. Yong, Pontryagin maximum principle for semilinear and quasilinear parabolic equations with pointwise state constraints, SIAM J. Control Optim., 33 (1995), 1857–1880. https://doi.org/10.1007/s10957-023-02346-6 doi: 10.1007/s10957-023-02346-6 [22] R. Kamocki, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 235 (2014), 94–104. https://doi.org/10.1016/j.amc.2014.02.086 doi: 10.1016/j.amc.2014.02.086 [23] R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668–1686. https://doi.org/10.1002/mma.2928 doi: 10.1002/mma.2928 [24] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. [25] M. I. Krastanov, N. K. Ribarska, T. Y. Tsachev, A Pontryagin maximum principle for infinite dimensional problems, SIAM J. Control Optim., 49 (2011), 2155–2182. https://doi.org/10.1137/100799009 doi: 10.1137/100799009 [26] X. Li, J. Yong, Optimal control theory for infinite dimensional systems, Birkhauser, 1995. https://doi.org/10.1007/978-1-4612-4260-4 [27] X. Li, S. Chow, Maximum principle of optimal control for functional differential systems, J. Optim. Theory Appl., 54 (1987), 335–360. https://doi.org/10.1007/BF00939438 doi: 10.1007/BF00939438 [28] P. Lin, J. Yong, Controlled singular Volterra integral equations and Pontryagin maximum principle, SIAM J. Control Optim., 58 (2020), 136–164. https://doi.org/10.1137/19M124602X doi: 10.1137/19M124602X [29] X. Liu, Q. Lü, H. Zhang, X. Zhang, Finite codimensionality method for infinite-dimensional optimization problems, arXiv, 2022. https://arXiv.org/abs/2102.00652 [30] X. Liu, Q. Lü, X. Zhang, Finite codimensional controllability and optimal control problems with endpoint state constraints, J. Math. Pures Appl., 138 (2020), 164–203. https://doi.org/10.1016/j.matpur.2020.03.004 doi: 10.1016/j.matpur.2020.03.004 [31] C. Lizama, Abstract nonlinear fractional evolution equations, In: A. Kochubei, Y. Luchko, Handbook of fractional calculus with applications: volume 2 fractional differential equations, Boston: De Gruyter, 2019,499–514. https://doi.org/10.1515/9783110571660-022 [32] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, An Introduction to Mathematical Models, Imperial College Press, 2010. [33] J. Moon, A necessary optimality condition for optimal control of caputo fractional evolution equations, IFAC-PapersOnLine, 56 (2023), 7480–7485. https://doi.org/10.1016/j.ifacol.2023.10.1299 doi: 10.1016/j.ifacol.2023.10.1299 [34] J. Moon, A Pontryagin maximum principle for terminal state-constrained optimal control problems of Volterra integral equations with singular kernels, AIMS Math., 8 (2023), 22924–22943. https://doi.org/10.3934/math.20231166 doi: 10.3934/math.20231166 [35] M. McAsey, L. Mou, A proof of a general maximum principle for optimal controls via a multiplier rule on metric space, J. Math. Anal. Appl., 337 (2008), 1072–1088. https://doi.org/10.1016/j.jmaa.2007.04.029 doi: 10.1016/j.jmaa.2007.04.029 [36] A. Pazy, Semigroups of linear operators and applications to partial differential equations, New York: Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1 [37] M. Radmanesh, M. J. Ebadi, A local mesh-less collocation method for solving a class of time-dependent fractional integral equations: Bolza fractional evolution equation, Eng. Anal. Bound. Elem., 113 (2020), 372–381. https://doi.org/10.1016/j.enganabound.2020.01.017 doi: 10.1016/j.enganabound.2020.01.017 [38] C. S. Sin, H. C. In, K. C. Kim, Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations, Adv. Differ. Equ., 2018 (2018), 61. https://doi.org/10.1186/s13662-018-1519-9 doi: 10.1186/s13662-018-1519-9 [39] C. Wang, S. Chen, Maximum principle for optimal control of some parabolic systems with two point boundary conditions, Numer. Func. Anal. Optim., 20 (1999), 163–174. https://doi.org/10.1080/01630569908816886 doi: 10.1080/01630569908816886 [40] G. Wang, L. Wang, State-constrained optimal control governed by non-well-posed parabolic differential equations, SIAM J. Control Optim., 40 (2002), 1517–1539. https://doi.org/10.1137/S0363012900377006 doi: 10.1137/S0363012900377006 [41] J. Wang, Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal.: Real World Appl., 12 (2011), 262–272. https://doi.org/10.1016/j.nonrwa.2010.06.013 doi: 10.1016/j.nonrwa.2010.06.013 [42] J. Wang, Y Zhou, W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Control Lett., 61 (2012), 472–476. https://doi.org/10.1016/j.sysconle.2011.12.009 doi: 10.1016/j.sysconle.2011.12.009 [43] H. Ye, J. Gas, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061 doi: 10.1016/j.jmaa.2006.05.061 [44] T. A. Yıldız, A. Jajarmi, B. Yıldız, D. Baleanu, New aspects of time fractional optimal control problems within operators with nonsingular kernel, Discrete Cont. Dyn. Syst. (Ser. S), 13 (2020), 407–428. https://doi.org/10.3934/dcdss.2020023 doi: 10.3934/dcdss.2020023 [45] J. Yong, Pontryagin maximum principle for semilinear second order elliptic partial differential equations and variational inequalities with state constraints, Differ. Integral Equ., 5 (1992), 1307–1334. https://doi.org/10.57262/die/1370875549 doi: 10.57262/die/1370875549 [46] X. Zhang, H. Li, C. Liu, Optimal control problem for the Cahn-Hilliard/Allen-Cahn equation with state constraint, Appl. Math. Optim., 82 (2020), 721–754. https://doi.org/10.1007/s00245-018-9546-1 doi: 10.1007/s00245-018-9546-1 [47] Y. Zhou, Fractional evolution equations and inclusions: analysis and control, Academic Press, 2016. https://doi.org/10.1016/C2015-0-00813-9 [48] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077. https://doi.org/10.1016/j.camwa.2009.06.026 doi: 10.1016/j.camwa.2009.06.026
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