Gain Scheduling-Inspired Control for Nonlinear Partial Differential Equations

2009 ◽  
Author(s):  
Antranik A. Siranosian ◽  
Miroslav Krstic ◽  
Andrey Smyshlyaev ◽  
Matt Bement
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Design Method ◽  
Nonlinear Partial Differential Equations ◽  
Gain Scheduling ◽  
System Control ◽  
Nonlinear Feedback ◽  
Partial Differential ◽  
System A ◽  
Shear Beam

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with a destabilizing in-domain nonlinearity is considered first. For this system a nonlinear feedback law based on gain scheduling is derived explicitly, and a statement of stability is presented for the closed-loop system. Control designs are then presented for a string and shear beam PDE, both with Kelvin-Voigt damping and potentially destabilizing free-end nonlinearities. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization-based design.

Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

2011 ◽  
Vol 133 (5) ◽  
Author(s):  
Antranik A. Siranosian ◽  
Miroslav Krstic ◽  
Andrey Smyshlyaev ◽  
Matt Bement
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Design Method ◽  
Nonlinear Partial Differential Equations ◽  
Gain Scheduling ◽  
System Control ◽  
Nonlinear Feedback ◽  
Partial Differential ◽  
System A ◽  
Shear Beam

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability, with an estimate of the region of attraction, is presented for the closed-loop system. Control designs (without proofs) are then presented for a string PDE and a shear beam PDE, both with Kelvin–Voigt (KV) damping and free-end nonlinearities of a potentially destabilizing kind. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization based design.

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

1996 ◽  
Vol 7 (6) ◽  
pp. 635-666 ◽  
Author(s):  
Gregory J. Reid ◽  
Allan D. Wittkopf ◽  
Alan Boulton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Original System ◽  
Polynomial Ideal ◽  
Partial Differential ◽  
System A ◽  
Integrability Conditions ◽  
Algebraic Operations

We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Wave ◽  
Symbolic Computation ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Wave Model ◽  
Simple Equation ◽  
Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

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