The optimal control of a new class of impulsive stochastic neutral evolution integro-differential equations with infinite delay. (English) Zbl 1353.93120
Summary: In this paper, we introduce the optimal control problems governed by a new class of impulsive stochastic partial neutral evolution equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, the analytic semigroup theory, fractional powers of closed operators, and suitable fixed point theorems, we prove an existence result of mild solutions for the control systems in the \(\alpha\)-norm without the assumptions of compactness. Next, we derive the existence conditions of optimal pairs of these systems. Finally, application to a nonlinear impulsive stochastic parabolic optimal control system is considered.
MSC:
| 93E20 | Optimal stochastic control |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93C10 | Nonlinear systems in control theory |
Keywords:
impulsive stochastic evolution integro-differential equations; optimal controls; infinite delay; analytic semigroup theory; fixed point theoremReferences:
| [1] | DOI: 10.1137/S0363012901391299 · Zbl 1037.49036 · doi:10.1137/S0363012901391299 |
| [2] | DOI: 10.1016/j.na.2014.01.019 · Zbl 1285.49017 · doi:10.1016/j.na.2014.01.019 |
| [3] | DOI: 10.14232/ejqtde.2009.1.67 · Zbl 1203.34116 · doi:10.14232/ejqtde.2009.1.67 |
| [4] | DOI: 10.1080/00207179.2015.1006683 · Zbl 1316.93119 · doi:10.1080/00207179.2015.1006683 |
| [5] | DOI: 10.1007/BF02843536 · Zbl 0934.45012 · doi:10.1007/BF02843536 |
| [6] | DOI: 10.1016/0362-546X(87)90092-7 · Zbl 0638.49004 · doi:10.1016/0362-546X(87)90092-7 |
| [7] | DOI: 10.1155/9789775945501 · doi:10.1155/9789775945501 |
| [8] | DOI: 10.1002/mana.19981890103 · Zbl 0896.47042 · doi:10.1002/mana.19981890103 |
| [9] | DOI: 10.1017/CBO9780511666223 · doi:10.1017/CBO9780511666223 |
| [10] | DOI: 10.1515/fca-2015-0007 · Zbl 1321.49007 · doi:10.1515/fca-2015-0007 |
| [11] | Friedman A, Partial differential equations (1969) |
| [12] | Gautam G.R., Malaya Journal of Matematik, 2 pp 428– (2014) |
| [13] | DOI: 10.1090/S0002-9939-2012-11613-2 · Zbl 1266.34101 · doi:10.1090/S0002-9939-2012-11613-2 |
| [14] | DOI: 10.1007/s10440-009-9546-x · Zbl 1202.60098 · doi:10.1007/s10440-009-9546-x |
| [15] | DOI: 10.1142/0906 · doi:10.1142/0906 |
| [16] | DOI: 10.11650/tjm.19.2015.3131 · Zbl 1357.49017 · doi:10.11650/tjm.19.2015.3131 |
| [17] | DOI: 10.1016/j.mcm.2009.12.006 · Zbl 1190.60045 · doi:10.1016/j.mcm.2009.12.006 |
| [18] | DOI: 10.1016/j.na.2006.07.013 · Zbl 1122.60057 · doi:10.1016/j.na.2006.07.013 |
| [19] | Mao X, Stochastic differential equations and their applications (1997) · Zbl 0892.60057 |
| [20] | Marle C.M, Measures et Probabilités (1974) |
| [21] | DOI: 10.1016/j.jmaa.2013.01.053 · Zbl 1260.93178 · doi:10.1016/j.jmaa.2013.01.053 |
| [22] | DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1 |
| [23] | DOI: 10.1016/j.amc.2012.12.084 · Zbl 1293.34019 · doi:10.1016/j.amc.2012.12.084 |
| [24] | DOI: 10.1002/mma.2720 · Zbl 1274.60212 · doi:10.1002/mma.2720 |
| [25] | DOI: 10.1007/s10957-010-9792-0 · Zbl 1241.34089 · doi:10.1007/s10957-010-9792-0 |
| [26] | DOI: 10.1016/j.cnsns.2013.05.015 · Zbl 1344.93019 · doi:10.1016/j.cnsns.2013.05.015 |
| [27] | DOI: 10.1016/j.jmaa.2009.02.002 · Zbl 1166.60037 · doi:10.1016/j.jmaa.2009.02.002 |
| [28] | DOI: 10.1016/j.cnsns.2012.04.020 · Zbl 1273.60077 · doi:10.1016/j.cnsns.2012.04.020 |
| [29] | DOI: 10.1142/S0217984910023359 · Zbl 1211.93026 · doi:10.1142/S0217984910023359 |
| [30] | DOI: 10.1016/j.na.2011.12.028 · Zbl 1243.34006 · doi:10.1016/j.na.2011.12.028 |
| [31] | DOI: 10.1016/j.na.2012.10.009 · Zbl 1261.34063 · doi:10.1016/j.na.2012.10.009 |
| [32] | DOI: 10.1142/9789812798664 · doi:10.1142/9789812798664 |
| [33] | DOI: 10.1007/BF01447659 · Zbl 0691.49014 · doi:10.1007/BF01447659 |
| [34] | DOI: 10.1007/s10957-011-9892-5 · Zbl 1357.49018 · doi:10.1007/s10957-011-9892-5 |
| [35] | DOI: 10.1007/s10957-012-0170-y · Zbl 1263.49038 · doi:10.1007/s10957-012-0170-y |
| [36] | DOI: 10.1080/02331930500530401 · Zbl 1101.45002 · doi:10.1080/02331930500530401 |
| [37] | Xiang X., Discrete and Continuous Dynamical Systems, 2005 pp 911– (2005) |
| [38] | Yan Z., Journal of Applied Analysis and Computation, 5 pp 329– (2015) |
| [39] | DOI: 10.1007/s13348-012-0063-2 · Zbl 1272.34107 · doi:10.1007/s13348-012-0063-2 |
| [40] | Yan Z., Electronic Journal of Differential Equations, 206 pp 1– (2013) |
| [41] | DOI: 10.1016/j.cnsns.2014.10.010 · Zbl 1339.34069 · doi:10.1016/j.cnsns.2014.10.010 |
| [42] | Zang Y., Boundary Value Problems, 193 pp 1– (2013) |
| [43] | DOI: 10.1080/00207179.2010.495161 · Zbl 1208.49004 · doi:10.1080/00207179.2010.495161 |
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.
