On Homogeneous Finite-Time Control for Linear Evolution Equation in Hilbert Space

Andrey Polyakov, Jean-Michel Coron, Lionel Rosier
2018 IEEE Transactions on Automatic Control  
Based on the notion of generalized homogeneity, a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. Method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with non-homogeneous operators. The design scheme
more » ... demonstrated for heat equation with the control input distributed on the segment [0, 1]. Recently [22], a generalized group of dilations was introduced for Banach spaces allowing us to extend all important properties (known before only for ODEs) to homogeneous evolution equations in Banach and Hilbert spaces. The generalized homogeneity can be established for many wellknown partial differential equations (PDEs) like heat, wave, Korteweg-de Vries, Saint-Venant, Burgers, Navier-Stocks and fast diffusion equations [22]. The finite-time control of PDEs is still the topic of intensive research [23], [24], [25], [26], [27]. It is related with controllability analysis of evolution systems [28] as well as with sliding mode control method [29], [30]. This technical note deals with homogeneous finite-time control design for plants modeled by linear evolution equations in Hilbert spaces. The design scheme follows the idea of the implicit Lyapunov function method [31], [32], which was recently linked with homogeneity theory [33]. Using This work is supported by ANR Project Finite4SoS (ANR 15-CE23-0007).
doi:10.1109/tac.2018.2797838 fatcat:3i3kr225djfffkfb2mhhvyvlvy