关于双曲型方程组最优边界控制问题的收费公路现象
On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems
摘要：
研究线性双曲型偏微分方程控制系统的最优边界控制问题。目标函数是二次的，并通过在有限时间间隔$(0,\, T)$上的积分给出，该时间间隔取决于解的边界迹。如果时间范围$T$足够大，则动态最优边界控制问题的解可以近似为稳态最优化问题的解。我们证明，如果将时间间隔$(0,\, T)$变换为固定间隔$(0,\, 1)$，则对于$T\rightarrow\infty$，逼近误差在Hilbert空间$L^2(0,\,1)$的范数意义下收敛于零，速率为$1/T$。此外，我们还表明，最优边界控制问题的整数约束的控制的收费公路现象发生。在这种情况下，稳态最优化问题也具有整数约束。如果$T$足够大，则具有整数约束的动态最优边界控制问题的每个解的整数部分等于静态问题的解的整数部分。对输气管道运行中的一个控制问题进行了数值验证。
英文摘要：
We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval $(0,\, T)$ that depends on the boundary traces of the solution. If the time horizon $T$ is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for $T\rightarrow\infty$ the approximation error converges to zero in the sense of the norm in the Hilbert space $L^2(0,\,1)$ with the rate $1/T$, if the time interval $(0,\, T)$ is transformed to the fixed interval $(0,\, 1)$. Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If $T$ is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.
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