AI 中文总结
该研究针对带边界控制的一维热方程，推导了$T\to0^+$时$H^{-1}$空间中最优$L^2$零控制代价的渐近式，确定的常数填补了相关下界与上界的差距。
AI 中文摘要
我们考虑区间$(0,L)$上的热方程，其一个端点处为齐次狄利克雷条件，另一个端点处为狄利克雷边界控制。若$C_{\mathrm H}(T,L)$表示初始数据属于$H^{-1}(0,L)$时的最优$L^2$零控制代价，我们证明当$T\to0^+$时，$C_{\mathrm H}(T,L)=\exp\left(\frac{\kappa_*L^2+o(1)}{T}\right)$，其中$\kappa_*=\frac{\Gamma(\frac14)^4}{8\pi^3}\simeq0.696601964842838$。常数$\kappa_*$与Dardé和Ervedoza（2019，ANPDE）得到的上界常数一致，该常数在原文中通过收敛级数表示，这一结果填补了Lissy（2015，JDE）得到的下界与Dardé和Ervedoza建立的上界之间的差距。
英文摘要
We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If \(C_{\mathrm H}(T,L)\) denotes the optimal \(L^2\) null-control cost for initial data in \(H^{-1}(0,L)\), we prove that \[ C_{\mathrm H}(T,L) = \exp\left(\frac{κ_*L^2+o(1)}{T}\right), \qquad κ_* = \frac{Γ(\frac14)^4}{8π^3} \simeq 0.696601964842838, \qquad T\to0^+. \] The constant \(κ_*\) coincides with the upper-bound constant obtained by Dardé and Ervedoza (2019, ANPDE), which was expressed there through a convergent series. This closes the gap between the lower bound obtained in by Lissy (2015, JDE) and the upper bound established by Dardé and Ervedoza.
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