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 $Y$.

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