Homotopic approach for turnpike and singularly perturbed optimal control problems

The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

A Random Access Protocol Approach to Stochastic Jumping Systems with Singular Perturbations

This paper focuses on the stochastic jumping systems with singular perturbation subject to a random access protocol. The key challenge with controller design issue of stochastic jumping systems is how to assess the coordination of communication orders. In this study, a joint Markov process is established, and a novel control law is proposed. In contrast with the existing methods, the developed controller is more general. Finally, a practical example is exhibited to show the effectiveness of the achieved theories.

Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \; \epsilon&gt;0 \tag{*} \label{0} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and the first order nonautonomous lattice systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \dot{u}_{m}+(Au)_{m}+\lambda _{m}u_{m}+f_{m}(u_{j}|j∈I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k}. \tag{**} \label{00} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>Under certain conditions, there are pullback attractors <inline-formula><tex-math id="M1">\begin{document}$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \{\mathcal{A}(t)\subset \ell ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> for systems (*)and (**), respectively. In this paper, we mainly consider the uppersemicontinuity of attractors <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{A}_{\epsilon }(t)\subset \ell^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula>, with respect to the coefficient <inline-formula><tex-math id="M5">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> of second derivative term under Hausdorff semidistance. First, we studythe relationship between <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M8">\begin{document}$ \epsilon \rightarrow 0^{+} $\end{document}</tex-math></inline-formula>. We construct a family of compact sets <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{A}_{0}(t)\subset \ell ^{2}\times \ell ^{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ t\in \mathbb{R} $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{A}(t) $\end{document}</tex-math></inline-formula> is naturally embedded into <inline-formula><tex-math id="M12">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> as the firstcomponent, and prove that <inline-formula><tex-math id="M13">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enter anyneighborhood of <inline-formula><tex-math id="M14">\begin{document}$ \mathcal{A}_{0}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M15">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> is small enough. Thenfor <inline-formula><tex-math id="M16">\begin{document}$ \epsilon _{0}&gt;0 $\end{document}</tex-math></inline-formula>, we prove that <inline-formula><tex-math id="M17">\begin{document}$ \mathcal{A}_{\epsilon }(t) $\end{document}</tex-math></inline-formula> can enterany neighborhood of <inline-formula><tex-math id="M18">\begin{document}$ \mathcal{A}_{\epsilon _{0}}(t) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \epsilon\rightarrow \epsilon _{0} $\end{document}</tex-math></inline-formula>. Finally, we consider the existence andexponentially attraction of the singleton pullback attractors of systems (*)-(**).</p>