A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

Zheng, Jiqiang; Zhu, Guchuan

doi:10.1007/s00498-020-00258-8

Cited by 11 publications

(10 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The navigation process of USVs is a nonlinear control and uncertain process, the navigation control of USVs in the future needs to combine with advanced control methods. Generally, the control theory is far from practical application, especially there are fewer advanced theoretical control methods that are applied for USVs, such as nonlinear control based on manifold space (Sakamoto, N. 2013) and partial differential equations control (Zheng, J. & Zhu, G. 2019) and so on.…”

Section: Discussionmentioning

confidence: 99%

The review unmanned surface vehicle path planning: Based on multi-modality constraint

et al. 2020

View full text Add to dashboard Cite

The essence of the path planning problems is multi-modality constraint. However, most of the current literature has not mentioned this issue. This paper introduces the research progress of path planning based on the multi-modality constraint. The path planning of multi-modality constraint research can be classified into three stages in terms of its basic ingredients (such as shape, kinematics and dynamics et al.): Route Planning, Trajectory Planning and Motion Planning. It then reviews the research methods and classical algorithms, especially those applied to the Unmanned Surface Vehicle (USV) in every stage. Finally, the paper points out some existing problems in every stage and suggestions for future research.

show abstract

Section: Discussionmentioning

confidence: 99%

The review unmanned surface vehicle path planning: Based on multi-modality constraint

et al. 2020

View full text Add to dashboard Cite

show abstract

“…In recent years, input-to-state stability of linear and non-linear partial differential equation systems -parabolic and hyperbolic -has attracted a great deal of research, both in the case of distributed inputs (entering in the domain) and in the typically more challenging case of boundary inputs (entering at the boundary of the domain of the considered pde). See [6], [16], [22], [23], [26], [31], [51], [50], [56], [57], [58], [10], [24], for instance. So far, the works establishing input-to-state stability of parabolic pde systems by far outnumber those on hyperbolic pde systems (like the port-Hamiltonian systems considered here).…”

Section: Introductionmentioning

confidence: 99%

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

Schmid

Zwart

2021

ESAIM: COCV

View full text Add to dashboard Cite

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \N$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed back into the system. What we show here is that the resulting nonlinear closed-loop system is input-to-state stable w.r.t.~square-integrable disturbance inputs. In particular, we obtain uniform input-to-state stability for systems of order $N=1$ and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary order $N \in \N$ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to $0$ of all solutions as $t \to \infty$. Applications are given to vibrating strings and beams.

show abstract

“…Rapid progress in the development of Input-to-State Stability (ISS) theory for systems modeled by Partial Differential Equations (PDEs) took place during the last decade (see for instance [3,7,8,9,12,14,16,20,21,22,28,29,30]). Researchers dealt with the two major problems that arise in the study of PDEs with inputs and do not appear in the study of finite-dimensional systems or delay systems: (i) the selection of the state norm (functional norms are not equivalent), and (ii) the presence of boundary inputs (the boundary inputs enter through unbounded operators).…”

Section: Introductionmentioning

confidence: 99%

“…However, there are differences between the obtained results with the most obvious difference being the lack of an ISS Lyapunov functional when maximum principles are used for the derivation of the ISS estimates. It should be noted that maximum principles have also been used in [30,31] in conjunction with Lyapunov functionals for the derivation of ISS estimates.…”

Section: Introductionmentioning

confidence: 99%

ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

Karafyllis

Krstić

2021

ESAIM: COCV

View full text Add to dashboard Cite

This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.

show abstract

A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances

Cited by 11 publications

References 34 publications

The review unmanned surface vehicle path planning: Based on multi-modality constraint

The review unmanned surface vehicle path planning: Based on multi-modality constraint

Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs

Contact Info

Product

Resources

About