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Singularities of stabilizing feedbacks. (English) Zbl 1024.93048
The paper deals with the stabilization problem of finite-dimensional control systems in the case of discontinuous feedback controls. The study is motivated by the fact that ODEs associated with such systems might not have any Carathéodory solution. Two main directions are identified for approaching this topic: (i) the use of arbitrary feedback controls, which requires adequate definitions for the generalized solutions of discontinuous ODEs, (ii) the use of a particular class of feedback controls, which allows considering solutions in the usual Carathéodory sense. Approach (i) relies on various definitions of generalized solutions found in the literature and their possible application to feedback stabilization. Approach (ii) exploits previous works of the same author, referring to patchy vector fields [with F. Ancona, ESAIM, Control Optim. Calc. Var. 4, 445-471 (1999; Zbl 0924.34058)] and directionally continuous vector fields [with G. Colombo, Boll. Unione Mat. Ital. (7) B 4, 295-311 (1990; Zbl 0709.34003)]. Some open problems related to approach (ii) are formulated in the last section of the paper.
MSC:
93D15Stabilization of systems by feedback
34A36Discontinuous equations
93C10Nonlinear control systems