Bilinear Controllability of Semilinear Partial Differential Equations

在受控分布参数系统的建模中，通常使用“加性”边界和内部局部分布控制。（这种控制的例子可以是热/传质过程中的源或放置在梁上的压电陶瓷驱动器。）就应用而言，这种控制似乎只能充分地模拟那些受控过程，这些过程不会因控制动作而改变其主要物理特性。它们更倾向于描述各种外部添加的“外来”来源和/或力量对当前过程的影响。然而，这一限制排除了大量新的和不太新的技术，例如，“智能材料”和许多生物医学、化学和核链式反应，它们能够在某些有目的的诱导条件下改变其主要参数（“催化剂”）。本提案的目的是通过引入和研究乘法（或“双线性”）控制来解决刚刚概述的半线性偏微分方程（PDEs）全局可控性背景下的问题。这些控制作为系数进入系统方程。因此，它们至少可以改变手头过程的一些主要参数，例如，光束的固有频率响应或化学反应的速率。（在前一种情况下，这可能是由嵌入的“智能”合金引起的，而在后一种情况下，则是由各种催化剂和/或反应成分机械混合的速度引起的。）本建议的重点是在双线性控制的框架下，发展一种新的方法来研究半线性反应-扩散-对流方程和波束方程的全局可控性。我们特别感兴趣的是这种乘法控制可能对高度非线性偏微分方程的可控性问题产生的影响，在这种情况下，经典的加性控制往往显得不够。

0204037Khapalov In the modeling of controlled distributed parameter systems "additive" boundary and internal locally distributed controls are typically used. (Examples of such controls can be a source in a heat/mass-transfer process or a piezoceramic actuator placed on a beam.) In terms of applications it appears that such controls can adequately model only those controlled processes that do not change their principal physical characteristics due to the control actions. They rather describe the effect of various externally added "alien" sources and/or forces on the process at hand. This limitation, however, excludes a vast array of new and not quite new technologies, such as, for example, "smart materials" and numerous biomedical, chemical and nuclear chain reactions, which are able to change their principal parameters under certain purposefully induced conditions ("catalysts").The intent of this proposal is to address the just-outlined issues in the context of global controllability of semilinear partial differential equations (PDEs) through the introduction and study of multiplicative (or "bilinear") controls. These controls enter the system equations as coefficients. Accordingly they can change at least some of the principal parameters of the process at hand, such as, for example, the natural frequency response of a beam or the rate of a chemical reaction. (In the former case this can be caused, e.g., by the embedded "smart" alloys and, in the latter case, by various catalysts and/or by the speed at which the reaction ingredients are mechanically mixed.) The focus of this proposal is on the development of a new methodology for the study of global controllability of the semilinear reaction-diffusion-convection equation and of the wave and beam equations in the framework of bilinear controls. We are particularly interested in the effect such multiplicative controls may have on the issue of controllability of highly nonlinear PDEs, in which case the classical additive controls often appear to be inadequate.