754 research outputs found
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 -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
lies within the control region or not. For each case, we
establish specific Carleman estimates. As a result, we achieve null
controllability in the first case and unique continuation and
approximate controllability in the second case
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