×
Jeong, Jin-Mun; Son, Sang-Jin

Time optimal control of semilinear control systems involving time delays. (English) Zbl 1326.49006

J. Optim. Theory Appl. 165, No. 3, 793-811 (2015).
In this article, the authors consider the infinite-dimensional time-optimal control problem to a target set governed by a semilinear retarded functional differential equation of parabolic type with unbounded principal operators in a Hilbert space. The control space is a real Banach space and the admissible set stands for a weakly compact subset in the control space. They use a fundamental solution that they build, along with an “easy consequence of the definition of real interpolation spaces.” Furthermore, a convergence theorem of time-optimal controls for the given semilinear retarded system to a point target is given.
Reviewer: Jafar Zafarani (Isfahan)

Cited in 24 Documents

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49J27 Existence theories for problems in abstract spaces
93C10 Nonlinear systems in control theory
35R10 Partial functional-differential equations

Keywords:

time-optimal control; semilinear retarded functional differential equation; target set; fundamental solution; real interpolation spaces
Cite Review PDF
Full Text: DOI

References:

[1] Nakagiri, S.: Optimal control of linear retarded systems in Banach spaces. J. Math. Anal. Appl. 120, 169-210 (1986) · Zbl 0603.49005 · doi:10.1016/0022-247X(86)90210-6
[2] Di Blasio, G., Kunisch, K., Sinestrari, E.: \[L^2\] L2-regularity for parabolic partial integro differential equations with delay in the highest-order derivatives. J. Math. Anal. Appl. 102, 38-57 (1984) · Zbl 0538.45007 · doi:10.1016/0022-247X(84)90200-2
[3] Jeong, J.M.: Retarded functional differential equations with \[L^1\] L1-valued controller. Funkc. Ekvacioj 36, 71-93 (1993) · Zbl 0791.35140
[4] Yong, J., Pan, L.L.: Quasi-linear parabolic partial differential equations with delays in the highest order partial derivatives. J. Aust. Math. Soc. 54, 174-203 (1993) · Zbl 0804.35139 · doi:10.1017/S1446788700037101
[5] Jeong, J.M., Kwun, Y.C., Park, J.Y.: Approximate controllability for semilinear retarded functional differential equations. J. Dyn. Control Syst. 5, 329-346 (1999) · Zbl 0962.93013 · doi:10.1023/A:1021714500075
[6] Mordukhovich, B.S., Wang, D., Wang, L.: Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions. Nonlinear Anal. 71, 2740-2749 (2009) · Zbl 1239.49021 · doi:10.1016/j.na.2009.06.022
[7] Mordukhovich, B.S., Wang, D., Wang, L.: Optimization of delay-differential inclusions in infinite dimensions. Pac. J. Optim. 6, 353-374 (2010) · Zbl 1357.49065
[8] Micu, S., Roventa, I., Tucsnak, M.: Time optimal boundary controls for the heat equation. J. Funct. Anal. 263, 25-49 (2012) · Zbl 1243.49005 · doi:10.1016/j.jfa.2012.04.009
[9] Krakowiak, A.: Time optimal control of retarded parabolic systems. IMA J. Math. Control Inf. 24, 357-369 (2007) · Zbl 1131.49004 · doi:10.1093/imamci/dnl027
[10] Fattorini, H.O.: Time and norm optimal controls: a survey of recent results and open problems. Acta Math. Sci. Ser. B Engl. Ed. 31, 2203-2218 (2011) · Zbl 1265.49028 · doi:10.1016/S0252-9602(11)60394-9
[11] Tanabe, H.: Fundamental solutions for linear retarded functional differential equations in Banach space. Funkc. Ekvacioj 35, 239-252 (1992)
[12] Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems amd Applications I. Springer, Berlin (1972) · Zbl 0223.35039 · doi:10.1007/978-3-642-65161-8
[13] Wang, P.K.C.: Optimal control of parabolic systems with boundary conditions involving time delay. SIAM J. Control 13, 274-293 (1975) · Zbl 0301.49009 · doi:10.1137/0313016
[14] Droniou, J., Raymond, J.P.: Optimal pointwise control of semilinear parabolic. Nonlinear Anal. 39, 135-156 (2000) · Zbl 0939.49005 · doi:10.1016/S0362-546X(98)00170-9
[15] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, North-Holland (1978) · Zbl 0387.46032
[16] Aubin, J.P.: Un théorème de compacité. C. R. Acad. Sci. 256, 5042-5044 (1963) · Zbl 0195.13002
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.