双对角混合n耦合PDE级联系统的递阶多级能量控制方法
A hierarchic multi-level energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDE's by a reduced number of controls
摘要：
本文研究抽象级联双曲系统的精确能控性/能观性问题。我们证明了在级联双对角系统的耦合算子的充要条件下，通过观测矢量态的最后一个分量，可以在适当的泛函空间中恢复其所有分量的初始能量.该方法是基于一个多层次的能量方法，其中涉及$n$ -削弱的能量水平。我们建立了这个结果的情况下，有界以及无界的对偶控制算子和假设下的$n-1$耦合运营商的次对角线上的系统的部分连续性。通过$p+1$观测，我们进一步将可观测性结果推广到混合双对角和非双对角$n+p$耦合级联系统.应用HUM方法，我们得到了$n$耦合双对角级联和$n+p$耦合混合级联系统的精确能控性结果。利用波算子的嬗变方法，我们证明了相应的热（分别为）薛定谔）多维级联系统对于彼此不相交的控制区和耦合区以及对于大于$2$的维度的$n \le 5$和对于一维情况下的任何$n \ge 2$的任何正时间都是零可控的。控制可以被定位在子域或边界上，并且在一维情况下，耦合系数可以被支持在域的任何非空子集中。
英文摘要：
This work is concerned with the exact controllability/observability of abstract cascade hyperbolic systems by a reduced number of controls/observations. We prove that the observation of the last component of the vector state allows to recover the initial energies of all of its components in suitable functional spaces under a necessary and sufficient condition on the coupling operators for cascade bi-diagonal systems. The approach is based on a multi-level energy method which involves $n$-levels of weakened energies. We establish this result for the case of bounded as well as unbounded dual control operators and under the hypotheses of partial coercivity of the $n-1$ coupling operators on the sub-diagonal of the system. We further extend our observability result to mixed bi-diagonal and non bi-diagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying the HUM method, we derive the corresponding exact controllability results for $n$-coupled bi-diagonal cascade and $n+p$-coupled mixed cascade systems. Using the transmutation method for the wave operator, we prove that the corresponding heat (resp. Schrödinger) multi-dimensional cascade systems are null-controllable for control regions and coupling regions which are disjoint from each other and for any positive time for $n \le 5$ for dimensions larger than $2$, and for any $n \ge 2$ in the one-dimensional case. The controls can be localized on a subdomain or on the boundary and in the one-dimensional case the coupling coefficients can be supported in any non-empty subset of the domain.
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