论文标题
通过分布式或边界致动的线性耦合反应扩散PDE的稳定
Stabilization of underactuated linear coupled reaction-diffusion PDEs via distributed or boundary actuation
论文作者
论文摘要
这项工作涉及级联（反应 - 扩散系统）中M抛物线部分微分方程（PDE）的线性均相系统的指数稳定，其中仅在内部或从右边界控制第一个状态，并且扩散系数在其内部受到控制。对于分布式控制案例，明确给出了比例类型的稳定控制。应用模态分解后，稳定定律基于对应于相对不稳定模式的普通微分方程（ODE）系统的转换，其中稳定定律的计算与任意大量的这些模式无关。这是通过递归解决广义的Sylvester方程来实现的。对于边界控制情况，在耦合矩阵（反应项）的适当条件下，提出的控制器是动态的。首先执行控制控制的变量的三角变化，这是一种动态扩展技术。然后，应用模态分解，然后进行ode系统的状态转换，必须稳定下来才能以建立动态定律的形式写入。对于分布式和边界控制系统，提出了一种建设性和可扩展的稳定算法，因为控制器的选择与不稳定模式的数量无关，并且仅依赖于反应项的稳定。目前的方法解决了不足的系统稳定问题时，在存在独特的扩散系数的情况下，该问题无法直接解决，类似于标量PDE情况。关键字：线性抛物线PDE系统，不足的系统，稳定，模态分解
This work concerns the exponential stabilization of underactuated linear homogeneous systems of m parabolic partial differential equations (PDEs) in cascade (reaction-diffusion systems), where only the first state is controlled either internally or from the right boundary and in which the diffusion coefficients are distinct. For the distributed control case, a proportional-type stabilizing control is given explicitly. After applying modal decomposition, the stabilizing law is based on a transformation for the ordinary differential equations (ODE) system corresponding to the comparatively unstable modes into a target one, where the calculation of the stabilization law is independent of the arbitrarily large number of these modes. This is achieved by solving generalized Sylvester equations recursively. For the boundary control case, under appropriate sufficient conditions on the coupling matrix (reaction term), the proposed controller is dynamic. A dynamic extension technique via trigonometric change of variables that places the control internally is first performed. Then, modal decomposition is applied followed by a state transformation of the ODE system, which must be stabilized in order to be written in a form where a dynamic law can be established. For both distributed and boundary control systems, a constructive and scalable stabilization algorithm is proposed, as the choice of the controller gains is independent of the number of unstable modes and only relies on the stabilization of the reaction term. The present approach solves the problem of stabilization of underactuated systems when in the presence of distinct diffusion coefficients, the problem is not directly solvable, similarly to the scalar PDE case. Keywords: Linear parabolic PDE systems, underactuated systems, stabilization, modal decomposition
