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1. Verifying Fundamental Solution Groups for Lossless Wave Equations via Stationary Action and Optimal Control

   https://doi.org/10.1007/s00245-020-09700-4  

   Dower, Peter M. ; McEneaney, William M. ( August 2020 , Applied Mathematics & Optimization)

   null (Ed.)

   A new optimal control based representation for stationary action trajectories is constructed by exploiting connections between semiconvexity, semiconcavity, and stationarity. This new representation is used to verify a known two-point boundary value problem characterization of stationary action. 

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2. Gyroscopic Balancer-Enhanced Motion Control of an Autonomous Bikebot

   https://doi.org/10.1115/1.4063014  

   Wang, Pengcheng ; Han, Feng ; Yi, Jingang ( October 2023 , Journal of Dynamic Systems, Measurement, and Control)

   Bikebot (i.e., bicycle-based robot) is a class of underactuated balance robotic systems that require simultaneous trajectory tracking and balance control tasks. We present a tracking and balance control design of an autonomous bikebot. The external-internal convertible structure of the bikebot dynamics is used to design a causal feedback control to achieve both the tracking and balance tasks. A balance equilibrium manifold is used to define and capture the platform balance profiles and coupled interaction with the trajectory tracking performance. To achieve fully autonomous navigation, a gyrobalancer actuation is integrated with the steering and velocity control for stationary platform balance and stationary-moving switching. Stability and convergence analyses are presented to guarantee the control performance. Extensive experiments are presented to validate and demonstrate the autonomous control design. We also compare the autonomous control performance with human riding experiments and similar action strategies are found between them.

    

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3. Min-max and stat game representations for nonlinear optimal control problems

   https://doi.org/10.23919/ACC55779.2023.10156263  

   Dower, Peter M. ; McEneaney, William M. ; Zheng, Y ( May 2023 , Proceedings of the American Control Conference)

   A finite horizon nonlinear optimal control problem is considered for which the associated Hamiltonian satisfies a uniform semiconcavity property with respect to its state and costate variables. It is shown that the value function for this optimal control problem is equivalent to the value of a min-max game, provided the time horizon considered is sufficiently short. This further reduces to maximization of a linear functional over a convex set. It is further proposed that the min-max game can be relaxed to a type of stat (stationary) game, in which no time horizon constraint is involved. 

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4. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

   https://doi.org/10.3934/eect.2022007  

   Triggiani, Roberto ; Wan, Xiang ( January 2022 , Evolution Equations and Control Theory)

   We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

    

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5. Non-stationary Reinforcement Learning under General Function Approximation

   Feng, Songtao and ; Yin, Ming ; Huang, Ruiquan ; Wang, Yu-Xiang ; Yang, Jing ; Liang, Yingbin ( July 2023 , Proceedings of Machine Learning Research)

   Krause, Andreas and (Ed.)

   General function approximation is a powerful tool to handle large state and action spaces in a broad range of reinforcement learning (RL) scenarios. However, theoretical understanding of non-stationary MDPs with general function approximation is still limited. In this paper, we make the first such an attempt. We first propose a new complexity metric called dynamic Bellman Eluder (DBE) dimension for non-stationary MDPs, which subsumes majority of existing tractable RL problems in static MDPs as well as non-stationary MDPs. Based on the proposed complexity metric, we propose a novel confidence-set based model-free algorithm called SW-OPEA, which features a sliding window mechanism and a new confidence set design for non-stationary MDPs. We then establish an upper bound on the dynamic regret for the proposed algorithm, and show that SW-OPEA is provably efficient as long as the variation budget is not significantly large. We further demonstrate via examples of non-stationary linear and tabular MDPs that our algorithm performs better in small variation budget scenario than the existing UCB-type algorithms. To the best of our knowledge, this is the first dynamic regret analysis in non-stationary MDPs with general function approximation. 

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