Boundary Control Of Two Types Of First-order Hyperbolic Equations

This paper mainly studies the stabilizability and controllability of two kinds of first order hyperbolic equations with special spatial memory.At the same time,the exponential stability and the finite time stability of the heat-like equation in the rectangular region are considered.Compared with the existing backstepping method,an improved transformation method is introduced,which adds Fredholm integral term on the basis of Volterra integral term,reduces the calculation amount and breaks through the limitation of too complex solution process.In this paper,a series of mathematical skills are used to obtain the exact solution of the kernel function through complex integral calculation.At the same time,the concrete form of the inverse transformation is derived,and the invertibility of the introduced transformation is proved.For the exact controllability of the first order hyperbolic equation with a midpoint memory term,a new transformation is introduced to transform the original control system into a target control system with precise controllability.On the basis of proving the existence of kernel function in forward and inverse transformation,according to the reversibility of transformation and the precise controllability of target control system,the precise controllability of original control system is finally proved.For the stability of the first order hyperbolic equation with non causality in space,the exact solution of kernel function is obtained by using new transformation and complex calculation.At the same time,the concrete form of inverse transformation is deduced,and the existence of kernel function and the invertibility of transformation are proved.In this paper,a feedback controller is designed,which proves that the closed-loop system is exponentially stable.For the problem of exponential stability and finite time stability of heat-like equation in rectangular region,firstly,the two-dimensional heat-like equation is transformed into one-dimensional heat equation by the expansion of characteristic function,then the existence of a series of kernel functions in the transformation is obtained by using Backstepping method.Finally,the exponential and finite time stabilities of the heat-like equations are obtained by combining the state and control estimates of a series of heat equations.