本文将基于模型的策略迭代方法推广到了分布式时滞系统的线性二次最优控制问题(LQR)的求解,证明了由该迭代方法得到的性能指标是递减的,且控制器收敛于最优控制器。This paper extends the model-based policy iteration method to the solution of the Linear Quadratic Regulator (LQR) problem for distributed delayed systems. It is demonstrated that the performance criterion obtained through this iterative method is monotonically decreasing, and the controller converges to the optimal controller.
混合整数最优控制问题(Mixed-Integer Optimal Control Problem, MIOCP)因其包含整数变量而更加复杂,但更能贴近实际应用需求。Sager等人(Math. Program. Vol. 133, 2012)提出了一种松弛–取整策略,将MICOP问题凸松弛为经典最优控制,再对凸松弛的最优解进行取整近似(Sum Up Rounding, SUR),得到原问题的近似解,并证明了近似误差为时间步长的同阶无穷小。然而,该近似误差估计的同阶无穷小的系数项是时间区间总长度的指数函数,当控制问题的时间区间较大时,这个误差可能会非常大。针对这一问题,本文对SUR策略进行改进,提出一个新的控制取整策略,证明了新控制策略的收敛性,并通过数值例子验证了本文的策略显著提高了MIOCP的求解精度。Mixed-Integer Optimal Control Problems (MIOCP) are more complex due to the inclusion of integer variables but are more aligned with practical application needs. Sager et al. (Math. Program. Vol. 133, 2012) proposed a relaxation-rounding strategy that convexly relaxes the MIOCP to a classical optimal control problem and then approximates the optimal solution of the convex relaxation (Sum Up Rounding, SUR) to obtain an approximate solution to the original problem, proving that the approximation error is of the same order as the infinitesimal of the time step. However, the coefficient of this infinitesimal error estimate is an exponential function of the total length of the time interval, and when the time interval of the control problem is large, this error can be very significant. To address this issue, this paper improves the SUR strategy and proposes a new control rounding strategy, proving the convergence of the new control strategy and verifying through numerical examples that it significantly improves the solution accuracy of MIOCP.
在本文中,我们研究了三组式趋化系统的齐次Neumann初始边界值问题,该系统描述了斑秃的时空动态。与常规的趋化模型相比,该系统的一个显著特征是CD8+ T细胞在CD4+ T细胞的帮助下以非线性方式增殖。我们通过Leray-Schauder不动点定理证明了该系统强解的存在性、唯一性和正则性,随后建立了最优控制系统,推导出了系统全局最优解的存在性,并在Banach空间中利用拉格朗日乘子定理研究最优控制问题的一阶必要最优性条件,最后,我们得到了拉格朗日乘子的正则性结果。In this paper, we investigate the problem of homogeneous Neumann initial boundary values for a three-group chemotaxis system that describes the spatiotemporal dynamics of alopecia areata. To compare with previous chemotaxis models, a distinctive feature of this system is that CD8+ T cells additionally proliferate in a nonlinear manner with the help of CD4+ T cells. We prove the existence, uniqueness and regularity of the strong solution of the system by the Leray-Schauder fixed point theorem, then establish the optimal control system, deduce the existence of the global optimal solution of the system, and use the Lagrange multiplier theorem in Banach space to study the first-order necessary optimality condition of the optimal control problem, and finally, we obtain the regularity result of the Lagrange multiplier.
