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Lipschitz regularity of solution map of control systems with multiple state constraints

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  • Consider a closed subset $K \subset \mathbb{R}^n$ and $f:[0,T]\times \mathbb{R}^n\times U \to \mathbb{R}^n$, where $U$ is a complete separable metric space. We associate to these data the control system under a state constraint \begin{equation*}\label{dm400} \left \{ \begin{array}{lll} x'(t) &=&f(t,x(t),u(t)), \; \; u(t)\in U \quad\; \mbox{ a.e. in }\; [0,T] \\ x(t) & \in & K \quad\; \mbox{ for all }\; t \in [0,T]\\ x(0)& = &x_0 . \end{array} \right. \end{equation*} When the boundary of $K$ is smooth, then an inward pointing condition guarantees that under standard assumptions on $f$ (measurable in $t$, Lipschitz in $x$, continuous in $u$) the sets of solutions to the above system depend on the initial state $x_0$ in a Lipschitz way. This follows from the so-called Neighboring Feasible Trajectories (NFT) theorems. Some recent counterexamples imply that NFT theorems are not valid when $f$ is discontinuous in time and $K$ is a finite intersection of sets with smooth boundaries, that is in the presence of multiple state constraints.
        In this paper we prove that for multiple state constraints the inward pointing condition yields local Lipschitz dependence of solution sets on the initial states from the interior of $K$. Furthermore we relax the usual inward pointing condition. The novelty of our approach lies in an application of a generalized inverse mapping theorem to investigate feasible solutions of control systems. Our results also imply a viability theorem without convexity of right-hand sides for initial states taken in the interior of $K$.
    Mathematics Subject Classification: Primary: 34A60, 34H05, 54C60, 93C15; Secondary: 49K15, 49K24.


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