Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control where the operator is bilinear with respect to the control and the state in reflexive, separable Banach spaces denoted and , respectively. The operator is nonlinear in , and given a positive real number , the set-valued map is defined in . Without making any assumptions about the convexity of , its values are taken to be non-empty closed, decomposable subsets of .
DOI: 10.1051/cocv/2014055
Keywords: Nonlinear infinite system, differential inclusion, bilinear control, closed-loop control, feedback law, a priori estimates, Willett and Wong’s lemma
@article{COCV_2015__21_4_989_0, author = {Cl\'erin, Jean-Marc}, title = {Existence of solutions to bilinear problems with a closed-loop control}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {989--1001}, publisher = {EDP-Sciences}, volume = {21}, number = {4}, year = {2015}, doi = {10.1051/cocv/2014055}, mrnumber = {3395752}, zbl = {1326.93062}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2014055/} }
TY - JOUR AU - Clérin, Jean-Marc TI - Existence of solutions to bilinear problems with a closed-loop control JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2015 SP - 989 EP - 1001 VL - 21 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2014055/ DO - 10.1051/cocv/2014055 LA - en ID - COCV_2015__21_4_989_0 ER -
%0 Journal Article %A Clérin, Jean-Marc %T Existence of solutions to bilinear problems with a closed-loop control %J ESAIM: Control, Optimisation and Calculus of Variations %D 2015 %P 989-1001 %V 21 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2014055/ %R 10.1051/cocv/2014055 %G en %F COCV_2015__21_4_989_0
Clérin, Jean-Marc. Existence of solutions to bilinear problems with a closed-loop control. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 989-1001. doi : 10.1051/cocv/2014055. http://www.numdam.org/articles/10.1051/cocv/2014055/
J.-P. Aubin and A. Cellina, Differential inclusions, Set-valued maps and viability theory. Vol. 264 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1984). | MR | Zbl
J.-P. Aubin and H. Frankowska, Set-valued analysis. Birkhäuser, Boston (2008). | MR | Zbl
V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces. Vol. 10 of Math. Appl. (East European Series). D. Reidel Publishing Co., Dordrecht, romanian edition (1986). | MR | Zbl
Extensions and selections of maps with decomposable values. Studia Math. 90 (1988) 69–86. | DOI | MR | Zbl
and ,H. Brézis, Analyse fonctionnelle. Théorie et applications. [Theory and applications]. Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris (1983). | MR | Zbl
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions. Vol. 580 of Lect. Notes Math. Springer-Verlag, Berlin (1977). | MR | Zbl
F. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). | MR | Zbl
Équations d’état bien posées en contrôle bilinéaire (well-posed state equations in bilinear control). Rev. Roumaine Math. Pures Appl. 56 (2011) 115–136. | MR | Zbl
,Analyse de sensibilité d’un problème de contrôle optimal bilinéaire. Ann. Mat. Blaise Pascal 19 (2012) 177–196. | DOI | Numdam | MR | Zbl
,On the existence of optimal controls for nonlinear infinite-dimensional systems. Czechoslovak Math. J. 48 (1998) 291–312. | DOI | MR | Zbl
, and ,Continuous selections for a class of nonconvex multivalued maps. Studia Math. 76 (1983) 163–174. | DOI | MR | Zbl
,I.M. Gel’fand and N.Ya. Vilenkin, Generalized functions. Vol. 4 of Applications of harmonic analysis. Translated by Amiel Feinstein. Academic Press, New York (1964). | MR
S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis: Theory. Vol. 419 of Math. Appl. Springer (1997). | MR | Zbl
Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach. SIAM J. Control Optim. 41 (2003) 1886–1900. | DOI | MR | Zbl
,J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars (1969). | MR | Zbl
Continuous selections 1. Ann. Math. 63 (1956) 361–382. | DOI | MR | Zbl
,Compact sets in the space . Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl
,Vibration of a master plate with attached masses using modal sampling method. J. Acoust. Soc. Am. 96 (1994) 235–245. | DOI
and ,On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69 (1965) 362–367. | DOI | MR | Zbl
and ,Y.-Y. Yu, Vibrations of elastic plates. Springer (1995).
Cited by Sources: