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In this paper, we establish the null controllability for system coupled by two backward stochastic parabolic equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.

Background and Aim Hearing impairment is a commonly reported chronic condition among older adults. Hearing impairment is significantly associated with reduced quality of life and diminished function status. This study aimed to investigate the association between hearing impairment and mental health among Chinese older adults, with a focus on exploring the moderating effects of social participation and exercise on this association. Methods The data of this study were obtained from the 2018 wave of Chinese Longitudinal Healthy Longevity Survey (CLHLS). We employed ordinary least squares regression models to analyze the effect of hearing impairment on mental health. Propensity score matching (PSM) and doubly robust estimation were employed to conduct robustness checks. Results Hearing impairment produced an adverse effect on Mini-Mental State Examination (MMSE) score (coefficient = −1.4073, p < 0.001), while it had a positive effect on depression score (coefficient = 0.8682, p < 0.001). Further analyses using PSM and doubly robust estimation reported similar results. Moreover, social participation (coefficient = 0.9424, p < 0.001) and exercise (coefficient = 0.7001, p < 0.01) moderated the association between hearing impairment and MMSE score. Social participation (coefficient = −0.5991, p > 0.05) and exercise (coefficient = 0.7806, p > 0.05) did not moderate the association between hearing impairment and depression score. Conclusion We provide robust evidence indicating that hearing impairment had significantly negative effects on the cognitive function and depression status of older adults. Furthermore, we find that social participation and exercise relieved the negative effect of hearing impairment on cognitive function.

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The main purpose of this paper is to overview some recent methods and results on controllability/observability problems for systems governed by partial differential equations. First, the authors review the theory for linear partial differential equations, including the iteration method for the null controllability of the time-invariant heat equation and the Rellich-type multiplier method for the exact controllability of the time-invariant wave equation, and especially a unified controllability/observability theory for parabolic and hyperbolic equations based on a global Carleman estimate. Then, the authors present sharp global controllability results for both semi-linear parabolic and hyperbolic equations, based on linearization approach, sharp observability estimates for the corresponding linearized systems and the fixed point argument. Finally, the authors survey the local null controllability result for a class of quasilinear parabolic equations based on the global Carleman estimate, and the local exact controllability result for general hyperbolic equations based on a new unbounded perturbation technique.