Bilinear Controllability of Semilinear Partial Differential Equations

0204037 Khapalov在受控分布参数系统的建模中，通常使用“添加剂”边界和内部局部分布控制。（这种控制的例子可以是热/质量传递过程中的源或放置在梁上的压电陶瓷致动器。在应用方面，似乎这样的控制可以充分地模拟只有那些控制过程，不改变其主要的物理特性，由于控制行动。它们所描述的是各种外部添加的“外来”来源和/或力量对当前进程的影响。然而，这种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、化学和核连锁反应，它们能够在某些有目的的诱导条件下改变它们的主要参数（“催化剂”）。本提案的目的是解决公正的-通过介绍和研究乘法（或“双线性”）控制，概述了半线性偏微分方程（PDE）全局可控性方面的问题。这些控制输入系统方程作为系数。因此，它们可以改变至少一些当前工艺的主要参数，例如，梁的固有频率响应或化学反应速率。(In在前一种情况下，例如，通过嵌入的“智能”合金，并且在后一种情况下，通过各种催化剂和/或通过反应成分机械混合的速度。该建议的重点是发展一种新的方法，用于研究半线性反应扩散对流方程和双线性控制框架中的波和梁方程的全局可控性。我们特别感兴趣的效果，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，在这种情况下，经典的添加剂控制往往显得不足。

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.