An analysis on the existence of mild solution and optimal control for semilinear thermoelastic system. (English) Zbl 1540.74037
This article is concerned with the optimal control problem of a semilinear thermoelastic system, in which the control term is entirely included in the thermal equation. Using the contraction mapping for the considered system, the authors study the existence and uniqueness of mild solutions. Under some conditions on the corresponding linear differential operators and assuming that the nonlinear term is Lipschitz-continuous, the authors prove that Lagrange’s system recognises at least one optimal control pair.
Reviewer: Song Jiang (Beijing)
MSC:
| 74F05 | Thermal effects in solid mechanics |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
References:
| [1] | Fleming, W. H., Future Directions in Control Theory - A Mathematical Perspective (1988), Philadelphia: SIAM, Philadelphia |
| [2] | Balakrishnan, A. V., Optimal control problems in Banach spaces, J. Soc. Ind. Appl. Math. Ser. A Control, 3, 152-180 (1965) · Zbl 0178.44601 · doi:10.1137/0303014 |
| [3] | Park, D. G.; Jeong, J. M.; Kang, W. K., Optimal problem for retarded semilinear differential equations, J. Korean Math. Soc, 36, 2, 317-332 (1999) · Zbl 0929.49002 |
| [4] | Klamka, J., Schauder’s fixed-point theorem in nonlinear controllability problems, Control Cybern. J, 29, 1, 153-165 (2000) · Zbl 1011.93001 |
| [5] | Klamka, J., Decision and Control, 162, Controllability and Minimum Energy Control, Studies in Systems (2019), Cham: Springer, Cham · Zbl 1403.93002 |
| [6] | Jeong, J. M.; Hwang, H. J., Optimal control problems for semilinear retarded functional differential equations, J. Optim. Theory Appl, 167, 1, 49-67 (2015) · Zbl 1326.49009 · doi:10.1007/s10957-015-0726-8 |
| [7] | Wang, J.; Zhou, Y.; Medved, M., On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay, J. Optim. Theory Appl, 152, 1, 31-50 (2012) · Zbl 1357.49018 · doi:10.1007/s10957-011-9892-5 |
| [8] | Wang, J.; Zhou, Y.; Wei, W., Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Syst. Control Lett, 61, 4, 472-476 (2012) · Zbl 1250.49035 · doi:10.1016/j.sysconle.2011.12.009 |
| [9] | Papageorgiou, N. S., Existence of optimal controls for nonlinear systems in Banach spaces, J. Optim. Theory Appl, 53, 3, 451-459 (1987) · Zbl 0596.49006 · doi:10.1007/BF00938949 |
| [10] | Papageorgiou, N. S., On the optimal control of strongly nonlinear evolution equations, J. Math. Anal. Appl, 164, 1, 83-103 (1992) · Zbl 0766.49010 · doi:10.1016/0022-247X(92)90146-5 |
| [11] | Papageorgiou, N. S., Optimal control of nonlinear evolution equations with nonmonotone nonlinearities, J. Optim. Theory Appl, 77, 3, 643-660 (1993) · Zbl 0796.49006 · doi:10.1007/BF00940454 |
| [12] | Johnson, M.; Vijayakumar, V., An investigation on the optimal control for Hilfer fractional neutral stochastic integrodifferential systems with infinite delay, Fractal Fract, 6, 10, 1-22 (2022) · doi:10.3390/fractalfract6100583 |
| [13] | Johnson, M.; Mohan Raja, M.; Vijayakumar, V.; Shukla, A.; Nisar, K. S.; Jahanshahi, H., Optimal control results for impulsive fractional delay integrodifferential equations of order \(####\) via sectorial operator, Nonlinear Anal. Model. Control, 28, 3, 468-490 (2023) · Zbl 1519.45002 |
| [14] | Johnson, M.; Vijayakumar, V., An analysis on the optimal control for fractional stochastic delay integrodifferential systems of order, Fractal Fract., 7, 4, 1-24 (2023) |
| [15] | Johnson, M.; Vijayakumar, V., Optimal control results for Sobolev-type fractional stochastic Volterra-Fredholm integrodifferential systems of order \(####\) via sectorial operators, Numer. Funct. Anal. Optim., 44, 6, 439-460 (2023) · Zbl 1521.49005 · doi:10.1080/01630563.2023.2180645 |
| [16] | Dayal, R.; Malik, M.; Abbas, S.; Debbouche, A., Optimal controls for second-order stochastic differential equations driven by mixed-fractional Brownian motion with impulses, Math. Methods Appl. Sci, 43, 7, 4107-4124 (2020) · Zbl 1448.49034 |
| [17] | Li, K.; Peng, J.; Gao, J., Controllability of nonlocal fractional differential systems of order \(####\) in Banach spaces, Rep. Math. Phys., 71, 1, 33-43 (2013) · Zbl 1285.93023 |
| [18] | Patel, R.; Shukla, A.; Jadon, S. S., Existence and optimal control problem for semilinear fractional order \(####\) control system, Math. Methods Appl. Sci., 1-12 (2020) · doi:10.1002/mma.6662 |
| [19] | Kumar, S., Mild solution and fractional optimal control of semilinear system with fixed delay, J. Optim. Theory Appl, 174, 1, 108-121 (2017) · Zbl 1432.49032 · doi:10.1007/s10957-015-0828-3 |
| [20] | Kumar, S., The solvability and fractional optimal control for semilinear stochastic systems, Cubo, 19, 3, 1-14 (2017) · Zbl 1442.49033 · doi:10.4067/S0719-06462017000300001 |
| [21] | Tomar, N. K.; Sukavanam, N., Exact controllability of a semilinear thermoelastic system with control solely in thermal equation, Numer. Funct. Anal. Optim, 29, 9-10, 1171-1179 (2008) · Zbl 1151.93007 · doi:10.1080/01630560802467109 |
| [22] | Mohan Raja, M.; Vijayakumar, V.; Shukla, A.; Nisar, K. S.; Baskonus, H. M., On the approximate controllability results for fractional integrodifferential systems of order \(####\) with sectorial operators, J. Comput. Appl. Math., 415, 12 (2022) · Zbl 1492.93024 |
| [23] | Shukla, A.; Sukavanam, N.; Pandey, D. N., 2015 Proceedings of the Conference on Control and its Applications, Approximate controllability of semilinear fractional control systems of order, 175-180 (2015) · Zbl 07875188 |
| [24] | Shukla, A.; Sukavanam, N.; Pandey, D. N., Complete controllability of semilinear stochastic systems with delay in both state and control, Math. Rep, 18, 2, 247-259 (2016) · Zbl 1389.34242 |
| [25] | Lasiecka, I.; Pokojovy, M.; Wan, X., Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff-Love plate, Nonlinear Anal. Real World Appl, 38, 184-221 (2017) · Zbl 1377.35025 · doi:10.1016/j.nonrwa.2017.04.001 |
| [26] | Shukla, A.; Patel, R., Existence and optimal control results for second-order semilinear system in Hilbert spaces, Circuits Syst. Signal Process, 40, 4246-4258 (2021) · Zbl 1508.49002 · doi:10.1007/s00034-021-01680-2 |
| [27] | Mesloub, S.; Aldosari, F., On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate, Adv. Differ. Equ., 2021, 24, 1-15 (2021) · Zbl 1485.35401 · doi:10.1186/s13662-020-03188-6 |
| [28] | Avalos, G., Exact controllability of a thermoelastic system with control in the thermal component only, Differ. Integral Equ., 13, 4-6, 613-630 (2000) · Zbl 0970.93002 |
| [29] | Avalos, G.; Lasiecka, I., Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29, 1, 155-182 (1998) · Zbl 0914.35139 · doi:10.1137/S0036141096300823 |
| [30] | Zeidler, E., Nonlinear Functional Analysis and its Applications, I (1986), New York: Springer-Verlag, New York · Zbl 0583.47050 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
