Optimal Actuator Location of the Norm Optimal Controls for Degenerate Parabolic Equations

This paper focuses on investigating the optimal actuator location for achieving minimum norm controls in the context of approximate controllability for degenerate parabolic equations. We propose a formulation of the optimization problem that encompasses both the actuator location and its associated minimum norm control. Specifically, we transform the problem into a two-person zero-sum game problem, resulting in the development of four equivalent formulations. Finally, we establish the crucial result that the solution to the relaxed optimization problem serves as an optimal actuator location for the classical problem

Null controllability of two kinds of coupled parabolic systems with switching control

The focus of this paper is on the null controllability of two kinds of coupled systems including both degenerate and non-degenerate equations with switching control. We first establish the observability inequality for measurable subsets in time for such coupled system, and then by the HUM method to obtain the null controllability. Next, we investigate the null controllability of such coupled system for segmented time intervals. Notably, these results are obtained through spectral inequalities rather than using the method of Carleman estimates. Such coupled systems with switching control, to the best of our knowledge, are among the first to discuss

Semi-Supervised Hierarchical Recurrent Graph Neural Network for City-Wide Parking Availability Prediction

The ability to predict city-wide parking availability is crucial for the successful development of Parking Guidance and Information (PGI) systems. Indeed, the effective prediction of city-wide parking availability can improve parking efficiency, help urban planning, and ultimately alleviate city congestion. However, it is a non-trivial task for predicting citywide parking availability because of three major challenges: 1) the non-Euclidean spatial autocorrelation among parking lots, 2) the dynamic temporal autocorrelation inside of and between parking lots, and 3) the scarcity of information about real-time parking availability obtained from real-time sensors (e.g., camera, ultrasonic sensor, and GPS). To this end, we propose Semi-supervised Hierarchical Recurrent Graph Neural Network (SHARE) for predicting city-wide parking availability. Specifically, we first propose a hierarchical graph convolution structure to model non-Euclidean spatial autocorrelation among parking lots. Along this line, a contextual graph convolution block and a soft clustering graph convolution block are respectively proposed to capture local and global spatial dependencies between parking lots. Additionally, we adopt a recurrent neural network to incorporate dynamic temporal dependencies of parking lots. Moreover, we propose a parking availability approximation module to estimate missing real-time parking availabilities from both spatial and temporal domain. Finally, experiments on two real-world datasets demonstrate the prediction performance of SHARE outperforms seven state-of-the-art baselines.Comment: 8 pages, 9 figures, AAAI-202

Observability inequalities for the backward stochastic evolution equations and their applications

The present article delves into the investigation of observability inequalities pertaining to backward stochastic evolution equations. We employ a combination of spectral inequalities, interpolation inequalities, and the telegraph series method as our primary tools to directly establish observability inequalities. Furthermore, we explore three specific equations as application examples: a stochastic degenerate equation, a stochastic fourth order parabolic equation and a stochastic heat equation. It is noteworthy that these equations can be rendered null controllability with only one control in the drift term to each system

Some controllability results of a class of N-dimensional parabolic equations with internal single-point degeneracy

This paper investigates the controllability of a class of $N$-dimensional degenerate parabolic equations with interior single-point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region $\omega_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\in\omega_0$ and unique continuation and approximate controllability in the second case $0\notin\omega_0$