Optimal control, well-posedness and sensitivity analysis for a class of generalized evolutionary systems
DOI:
https://doi.org/10.12775/TMNA.2023.036Keywords
Optimal control, well-posedness, sensitivity analysis, fractional evolution inclusion, mixed variational-hemivariational inequalityAbstract
In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective of this paper is four fold. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set to the FDVHVI. The second aim is to study the optimal control problems described by the FDVHVI. The third goal is to establish the well-posedness results of the FDVHVI, including the existence, uniqueness, and stability. Furthermore, the sensitivity analysis of a perturbed problem associated to the FDVHVI with respect to the initial state and the two parameters is also obtained. Finally, a comprehensive fractional model is given to illustrate the validity of our main results.References
X.J. Chen and Z.Y. Wang, Differential variational inequality approach to dynamic games with shared constraints, Math. Program. 146 (2014), 379–408.
C. Christof, Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities, SIAM J. Control Optim. 57 (2019),no. 1, 192–218.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), 519–537.
L. Gasińsk and N.S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL, 2006.
S.C. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I, Theory, Kluwer Academic Publishers, Dordrecht Boston, London, 1997.
M. Kamenskiı̆, V. Obukhovskiı̆ ND P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Space, Walter de Gruyter, Berlin, 2001.
T.D. Ke, N.V. Loi and V. Obukhovskiı̆, Decay solutions for a class of fractional differential variational inequalities, Fract. Calc. Appl. Anal. 18 (2018), 531–553.
A.A. Kilbas, H.M. Srivastav and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elservier Science B.V., Amsterdam, 2006.
X.W. Li, Y.X. Li, Z.H. Liu and J. Li, Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions, Fract. Calc. Appl. Anal. 21 (2018), no. 6, 1439–1470.
X.W. Li and Z.H. Liu, Sensitivity analysis of optimal control problems described by differential hemivariational inequalities, SIAM J. Control Optim. 56 (2018), no. 5, 3569–3597.
X.Y. Liu and Z.H. Liu, On the “bang-bang” principle for a class of fractional semilinear evolution inclusions, Proc. Roy. Soc. Edinburgh Sect. A Math. 144 (2014), no. 2, 333–349.
Y.J. Liu, Z.H. Liu and N.S. Papageorgiou, Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities, J. Differential Equations 342 (2023), 559–595.
Z.H. Liu, N.V. Loi, V. Obukhovskiı̆, Existence and global bifurcation of periodic solutions to a class of differential variational inequalities, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013).
Z.H. Liu, S. Migórski and S.D. Zeng, Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces, J. Differential Equations 263 (2017), 3989–4006.
Z.H. Liu, D. Motreanu and S.D. Zeng, On the well-posedness of differential mixed quasivariational-inequalities, Topol. Methods Nonlinear Anal. 51 (2018), 135–150.
Z.H. Liu, D. Motreanu and S.D. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim. 31(2) (2021), 1158–1183.
Z.H. Liu and N.S. Papageorgiou, Double phase Dirichlet problems with unilateral constraints, J. Differential Equations 316 (2022), no. 15, 249–269.
Z.H. Liu, S.D. Zeng and D. Motreanu, Evolutionary problems driven by variational inequalities, J. Differential Equations 260 (2016), 6787–6799.
Z.H. Liu, S.D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal. 7 (2018), no. 4, 571–586.
N.V. Loi, T.D. Ke, V. Obukhovskiı̆ and P. Zecca, Topological methods for some classes of differential variational inequalities, J. Nonlinear Convex Anal. 17 (2016), 403–419.
S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics, vol. 26, Springer, New York, 2013.
S. Migórski and S.D. Zeng, A class of differential hemivariational inequalities in Banach spaces, J. Global Optim. 72 (2018), 761–779.
S. Migórski and S.D. Zeng, A class of generalized evolutionary problems driven by variational inequalities and fractional operators, Set-Valued Var. Anal. 27 (2019), 949–970.
J.S. Pang and D.E. Stewart, Differential variational inequalities, Math. Program. 113 (2008), 345–424.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer–Verlag, New York, 1983.
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
S.D. Zeng, Z.H. Liu and S. Migórski, A class of fractional differential hemivariational inequalities with application to contact problem, Z. Angew. Math. Phys. 69 (2018), DOI:10.1007/s00033-018-0929-6.
S.D. Zeng, S. Migórski and Z.H. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31 (2021), no. 4, 2829–2862.
J. Zhao, J. Chen and Z.H. Liu, Second order evolutionary problems driven by mixed quasi-variational chemivariational inequalities, Commun. Nonlinear Sci. Numer. Simulat. 120 (2023), 107192 ,14 pp.
J. Zhao, Z.H. Liu, E. Vilches, C.F. Wen and J-C Yao, Optimal control of an evolution hemivariational inequality involving history-dependent operators, Commun. Nonlinear Sci. Numer. Simulat. 103 (2021), 105992, 17 pp.
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.
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