混合时滞中立型系统的稳定性与可镇定性
Stability and stabilizability of mixed retarded-neutral type systems
摘要：
当中立项允许奇异时，我们分析了混合时滞中立型系统的稳定性和可镇定性。考虑系统在希尔伯特空间中的算子模型，我们感兴趣的是当存在一个实部趋近于零的特征值序列时的临界情形。在这种情况下，指数稳定性是不可能的，我们正在研究强渐近稳定性属性。混合滞后-中性型系统的谱行为不允许直接应用滞后系统的方法或中性型系统的方法进行稳定性分析。本文结合两种技巧得到了渐近非指数稳定的条件：有限维不变子空间的Riesz基的存在性和预解式在状态空间的一种特殊分解的某些子空间中的有界性。对于不稳定系统，所引入的技术允许分析混合延迟中立型系统的正则强可镇定性的概念。本文推广了R. Rabah，G.M. Sklyar，A.V. Rezounenko在[J. Diff. Equat.，214（2005），No.2，391-428]以及来自[J. Diff. Equat.，245（2008），No. 3，569-593]。
英文摘要：
We analyze the stability and stabilizability properties of mixed retarded-neutral type systems when the neutral term is allowed to be singular. Considering an operator model of the system in a Hilbert space we are interesting in the critical case when there exists a sequence of eigenvalues with real parts approaching to zero. In this case the exponential stability is not possible and we are studying the strong asymptotic stability property. The behavior of spectra of mixed retarded-neutral type systems does not allow to apply directly neither methods of retarded system nor the approach of neutral type systems for analysis of stability. In this paper two technics are combined to get the conditions of asymptotic non-exponential stability: the existence of a Riesz basis of invariant finite-dimensional subspaces and the boundedness of the resolvent in some subspaces of a special decomposition of the state space. For unstable systems the technics introduced allow to analyze the concept of regular strong stabilizability for mixed retarded-neutral type systems. The present paper extends the results by R. Rabah, G.M. Sklyar, A.V. Rezounenko on stability obtained in [J. Diff. Equat., 214(2005), No. 2, 391-428] and on stabilizability from [J. Diff. Equat., 245(2008), No. 3, 569-593].
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