High-Gain Observer Design for a Class of Quasi-Linear Integro-Differential Hyperbolic Systems—Application to an Epidemic Model

Abstract

This work addresses the problem of High-Gain Observer design for a class of quasi-linear hyperbolic systems (with one characteristic velocity), possibly including nonlocal terms, making them systems of Partial Integro-Differential Equations. The design relies on distributed measurement of a part of the state vector. The observer is presented and discussed and the exponential stability in the C 1 spatial norm of the origin for the error system is fully established via Lyapunov-based analysis. Its use is illustrated via an application to an age-dependent SIR epidemic model.

Contributions to the Problem of High-Gain Observer Design for Hyperbolic Systems

“…This is the reason why we call this approach direct. This result slightly generalizes [20] and [19] in the sense that it considers also the case of a velocity function of nonlocal nature.…”

“…Then, in the new coordinates, system is written in the general form (1a) we considered here, with constant velocities, namely, Λ [ξ 1 ] = I n and with its nonlinear source term having the form F [ξ (t)](x) := F (x, ξ (t)(x)), containing also nonlinear nonlocal terms, more explicitly, some integral terms of Volterra type and boundary terms. For the exact form of these mappings that we derive after the transformation (4), the reader can refer to [20].…”

“…The proof is included in the Appendix. A slightly different proof for a local velocity function has appeared in [20].…”

“…The proposed approach relies on Lyapunov analysis for spaces of higher regularity and the introduction of an infinite-dimensional state transformation. The direct observer design has already partially appeared in [20], without considering velocity functions of nonlocal nature. The indirect one is inspired by our previous work for semilinear parabolic systems [21] and for the case of quasilinear strictly hyperbolic systems, as the ones considered here, it has not appeared before.…”

Output Feedback Control of a Cascade System of Linear Korteweg--de Vries Equations

“…Here, we aim at observing the full state of a system of KdV equations written in a cascade form and finally controlling it, by considering a single observation. Observer design for nonlinear systems of partial differential equations written in such a form, based on the well-known high-gain methodology, has been considered, for instance, in [17,16,18], in the framework of first-order hyperbolic systems, extending results for finite-dimensional systems [15]. A similar form considered here, in its linearized version, allows an observer design, which relies on a choice of a sufficiently large parameter in its equations, while appropriate choice of the latter leads simultaneously to the closed-loop output feedback stabilization.…”