Wave Equation With Cone-Bounded Control Laws

• Published in: IEEE transactions on automatic control Vol. 61; no. 11; pp. 3452 - 3463
• Main Authors: Prieur, Christophe, Tarbouriech, Sophie, Gomes da Silva, João M.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Aerospace electronics; Asymptotic stability; Automatic; Boundary conditions; Closed loop systems; Control systems; Dynamics; Engineering Sciences; Infinite dimensional systems; Mathematical models; Nonlinear differential equations; nonlinear partial differential equations; Nonlinear programming; Nonlinearity; Partial differential equations; Propagation; saturated controls; Stability; Tuning; Wave equations
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2016.2519759

This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. Two kinds of control are considered: a distributed action and a boundary control. It is supposed that the control signal is subject to a cone-bounded nonlinearity. This kind of feedback laws includes (but is not restricted to) saturating inputs. By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups techniques. Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closed-loop system is proven by Lyapunov techniques. Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations.