Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions

Abstract

This paper addresses the boundary control problem of fluid transport in a Poiseuille flow taking the actuator dynamics into account. More precisely, sufficient stability conditions are derived to guarantee the exponential stability of a linear hyperbolic differential equation system subject to nonlinear quadratic dynamic boundary conditions by means of Lyapunov based techniques. Then, convex optimization problems in terms of linear matrix inequality constraints are derived to either estimate the closed-loop stability region or synthesize a robust control law ensuring the local closed-loop stability while estimating an admissible set of initial states. The proposed results are then applied to application-oriented examples to illustrate local stability and stabilization tools.

Feedback theory approach to positivity and stability of evolution equations

“…By using the positivity of MD λ , we have (I − MD λ ) −1 ≥ 0. Thanks to (10) and (11), we obtain λ ∈ ρ(A) and…”

“…As an application, we present an important class of systems called hyperbolic systems. We mention that there is a huge literature dealing with linear and non-linear hyperbolic systems (see e.g [9,3,4,10,11,14,22,23,24,12] and references therein). In particular, we are interested in the following linear hyperbolic partial differential equation with positive boundary conditions…”

Regional stability analysis of impulsive switched nonlinear systems with multiple equilibrium points

“…On the other hand, significant progress has been made in the theory of regional stability [22‐27]. The problem of regional stability analysis of two‐dimensional nonlinear polynomial systems represented by the Fornasini–Marchesini second state‐space model was addressed in Osowsky et al [22].…”

“…Silva et al [23] studied regional stability of a continuous‐time plant controlled by a discrete‐time dynamic output feedback control law with saturation constraints under aperiodic sampling. A linear matrix inequality was proposed for assessing the regional robust exponential stability of the closed‐loop system in Caldeira et al [24]. In Magni et al [25], the concept of regional input‐to‐state stability was introduced and studied.…”

Estimating the boundary of the region of attraction of Lotka–Volterra system with time delays

“…For such systems, it is of mandatory importance not only to determine whether the origin of the state space is locally asymptotically stable but also to ensure that the operative range is included into the convergence region of the equilibrium. Over the years, several papers have focused on the estimate of the RA of the zero-equilibrium point of quadratic system (Caldeira et al, 2018;Chen et al, 2013;Chesi et al, 2005;Chiang et al, 1988;Genesio & Vicino, 1984;Li-Ya et al, 2019;Merola et al, 2017;Tesi et al, 1996;Xu et al, 2020). In Genesio and Vicino (1984), a Lyapunov-based procedure is proposed to compute an ellipsoidal estimate of the RA of a quadratic system.…”