Volterra系统的能观性和能控性研究

We are concerned with observability and controllability of Volterra systems in infinite-dimensional Banach or Hilbert spaces. Observability and controllability play a key role in modern mathematical control theory and received much attention in the past three decades. We noticed that most existing research is restricted to Cauchy systems. However, owing to the viscoelasticity and thermoviscoelasticity of the material, many mathematical models from physical and engineering problems are reduced to Volterra systems and Cauchy systems correspond to the special case when the kernel function a(t)=0. For the general case, related problems become much more complicated and the results holding for Cauchy systems can not be generalized naturally to Volterra systems. Therefore, this project is meaningful from both the mathematical and the engineering point of view.

本项目拟研究无限维Banach空间或者Hilbert空间中Volterra线性系统的能观性和能控性。能观性与能控性是现代数学控制理论中两个核心的研究内容。我们注意到，相关的研究多限于Cauchy系统。然而，很多工程实际问题中往往都带有由粘弹性或者热粘弹性所导致的时滞，相应的数学模型归结为无限维状态空间中的Volterra方程和Volterra系统。Cauchy系统对应于Volterra系统核函数恒为零的特殊情形，我们已有的初步研究表明，核函数的出现使得有关的问题变得更复杂，很多对于Cauchy系统成立的结果并不能自然地推广到Volterra系统。无论从数学还是工程背景的角度来看，本项目都有重要的意义。

本项目《Volterra 系统的能观性和能控性研究》（基金号：11226241）自2012年11月获得资助以来，进展顺利。在基金资助下我们共完成论文三篇，其中一篇已发表在应用数学领域知名国际期刊Journal of Mathematical Analysis and Applications （简称JMAA）上；一篇发表在工程数学领域很好的杂志Journal of Engineering Mathematics (简称JEM)上，两篇文章皆为SCI源刊；另有一篇尚在审稿中。已发表的两篇文章研究了Volterra系统当核函数为零的特殊情形，研究了系统的解析解，我们认为这对于进一步的能观能控性研究是必要的。我们将理论结论应用到了热能工程领域中的换热器方程并与该领域的工程师合作，得到了非常好的结果。