CONVERGENCE RATES TO ASYMPTOTIC PROFILE FOR SOLUTIONS OF QUASILINEAR HYPERBOLIC EQUATIONS WITH LINEAR DAMPING

2011 ◽  
Vol 08 (01) ◽  
pp. 115-129 ◽  
Author(s):  
SHIFENG GENG
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Initial Data ◽  
Open Problem ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Asymptotic Profile ◽  
Linear Damping ◽  
Better Than

This paper is concerned with the asymptotic behavior of the solution of quasilinear hyperbolic equations with linear damping. The main novelty lies in the following observation: If we suitably choose the initial data of the corresponding parabolic equation, then the solution Ψ = Ψ(x, t) of the parabolic equation served as the new asymptotic profile satisfies ‖(V-Ψ, (V-Ψ)x, (V-Ψ)t)(t)‖L∞ = O(1)(t-2, t-5/2, t-3). The convergence rates of the new profile Ψ are better than that obtained by H.-J. Zhao (2000, J. Differential Equations167, 467–494), and we need none of the additional technical assumptions (H1) and (H2) therein. Therefore, we answer an open problem posed by Nishihara (1997, J. Differential Equations133, 384–395).

Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping

2015 ◽  
Vol 145 (5) ◽  
pp. 959-978 ◽  
Author(s):  
Shifeng Geng ◽  
Lina Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Nonlinear Damping ◽  
Boundary Effects ◽  
Time Behaviour ◽  
Quasilinear Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Quarter Plane

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

scholarly journals Reinforcement learning versus swarm intelligence for autonomous multi-HAPS coordination

2021 ◽  
Vol 3 (6) ◽  
Author(s):  
Ogbonnaya Anicho ◽  
Philip B. Charlesworth ◽  
Gurvinder S. Baicher ◽  
Atulya K. Nagar
Keyword(s):  
Reinforcement Learning ◽  
State Space ◽  
Swarm Intelligence ◽  
Performance Indicators ◽  
Convergence Rates ◽  
Tuning Parameters ◽  
Continuous State Space ◽  
Continuous State ◽  
User Coverage ◽  
Better Than

AbstractThis work analyses the performance of Reinforcement Learning (RL) versus Swarm Intelligence (SI) for coordinating multiple unmanned High Altitude Platform Stations (HAPS) for communications area coverage. It builds upon previous work which looked at various elements of both algorithms. The main aim of this paper is to address the continuous state-space challenge within this work by using partitioning to manage the high dimensionality problem. This enabled comparing the performance of the classical cases of both RL and SI establishing a baseline for future comparisons of improved versions. From previous work, SI was observed to perform better across various key performance indicators. However, after tuning parameters and empirically choosing suitable partitioning ratio for the RL state space, it was observed that the SI algorithm still maintained superior coordination capability by achieving higher mean overall user coverage (about 20% better than the RL algorithm), in addition to faster convergence rates. Though the RL technique showed better average peak user coverage, the unpredictable coverage dip was a key weakness, making SI a more suitable algorithm within the context of this work.

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