Existence of solutions to bilinear problems with a closed-loop control

Clérin, Jean-Marc

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 989-1001.

Here we prove the existence of solutions to nonlinear differential inclusion problems with closed-loop control $ẓ+A\left(z\right)=B(u,z)+f,u\in U(t,z),z\left(0\right)=z^0$ where the operator $B$ is bilinear with respect to the control $u$ and the state $z$ in reflexive, separable Banach spaces denoted $Y$ and $V$, respectively. The operator $A$ is nonlinear in $V$, and given a positive real number $T$, the set-valued map $U$ is defined in $[0,T]\times V$. Without making any assumptions about the convexity of $U$, its values are taken to be non-empty closed, decomposable subsets of $Y$.

Classification : 34A60, 35A01, 35G20, 93B52
Mots clés : Nonlinear infinite system, differential inclusion, bilinear control, closed-loop control, feedback law, a priori estimates, Willett and Wong’s lemma

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

Alberto Bressan and Giovanni Colombo, Extensions and selections of maps with decomposable values. Studia Math. 90 (1988) 69–86. | DOI | MR | Zbl

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

J.-M. Clérin, 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

J.-M. Clérin, Analyse de sensibilité d’un problème de contrôle optimal bilinéaire. Ann. Mat. Blaise Pascal 19 (2012) 177–196. | DOI | Numdam | MR | Zbl

A. Fiacca, N.S. Papageorgiou and F. Papalini, On the existence of optimal controls for nonlinear infinite-dimensional systems. Czechoslovak Math. J. 48 (1998) 291–312. | DOI | MR | Zbl

A. Fryszkowski, 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

A.Y. Khapalov, 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

E. Michael, Continuous selections 1. Ann. Math. 63 (1956) 361–382. | DOI | MR | Zbl

J. Simon, Compact sets in the space $L^p(0,T;B)$. Ann. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

D. Trentin and J.-L. Guyaner, Vibration of a master plate with attached masses using modal sampling method. J. Acoust. Soc. Am. 96 (1994) 235–245. | DOI

D. Willett and J.S.W. Wong, On the discrete analogues of some generalizations of Gronwall’s inequality. Monatsh. Math. 69 (1965) 362–367. | DOI | MR | Zbl

Y.-Y. Yu, Vibrations of elastic plates. Springer (1995).