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

Clérin, Jean-Marc¹

¹ Université Paris-Sorbonne (Paris IV), 10 rue Molitor, 75016 Paris, France.

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

Résumé

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 $Y$.

Détail

Reçu le : 2013-06-13
DOI : 10.1051/cocv/2014055

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