PDP Boundary Control for a Class of 2 × 2 Hyperbolic Systems with Variable Coefficients
Different from proportional feedback or PI feedback,a new kind of con-troller—A linear combination of position feedback and time-delayed position feed-back is proposed to stabilize a class of 2 × 2 hyperbolic conservation law systems with variable coefficients.Firstly,considered that the time-lag term can be described by the solution of the initial value problem of the first-order transportation equation,the system is transformed into a cascaded PDE-PDE closed-loop system,which con-sists of four partial differential equations.Furthermore,it can be rewritten in the form of an abstract evolution equation.Secondly,the well-posedness of the system is proven by using operator semigroup theory.Thirdly,the characteristic equation of the system operator,which is a transcendental equation with three exponential terms,is established,and the delay-independent stability and instability conclusions are obtained according to the theorem of the distribution of the zeros of exponential polynomials.Fourthly,the exponential stability of the closed-loop system is analyzed by constructing an appropriate weighted Lyapunov function,and the parameter dis-sipation condition related to the time delay value is established.Finally,the effective-ness of the time-delayed controller and the feasibility of the parameters are verified by numerical examples.
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