scholarly journals On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control

2020 ◽  
Vol 26 ◽  
pp. 23
Author(s):  
Mathias Dus ◽  
Francesco Ferrante ◽  
Christophe Prieur
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Weak Solutions ◽  
Hyperbolic Systems ◽  
Hyperbolic Partial Differential Equations ◽  
Lyapunov Analysis ◽  
Well Posedness ◽  
Nonlinear Semigroup ◽  
Region Of Attraction ◽  
Semilinear Hyperbolic Systems

This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L2([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L∞ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.

scholarly journals Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable

1990 ◽  
Vol 33 (3) ◽  
pp. 443-460 ◽  
Author(s):  
F. Demengel ◽  
J. Rauch
Keyword(s):  
Initial Data ◽  
Weak Convergence ◽  
Radon Measure ◽  
Space Variable ◽  
Hyperbolic Systems ◽  
Superposition Principle ◽  
Nonlinear Superposition ◽  
Image Position ◽  
Semilinear Hyperbolic Systems ◽  
Characteristic Coordinates

We study systems which in characteristic coordinates have the formwhere A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.Simple weak converge of the initial data does not imply weak convergence of the solutions.

Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions

2016 ◽  
Vol 146 (5) ◽  
pp. 1047-1080 ◽  
Author(s):  
Gilbert Peralta ◽  
Georg Propst
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Hyperbolic Systems ◽  
Well Posedness ◽  
Dynamic Boundary Conditions ◽  
Weak Formulation ◽  
Dynamic Boundary ◽  
Additional Regularity ◽  
The Waves ◽  
Interior Points

We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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