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Reachability of weakly nonlinear systems using Carleman linearization [article]

Marcelo Forets, Christian Schilling
2021 arXiv   pre-print
Using a global error bound for the Carleman linearization abstraction, we are able to describe the full set of behaviors of the system for sets of initial conditions and in dense time.  ...  The novelty of the approach is that we employ a particular embedding (Carleman linearization) to leverage recent advances of high-dimensional reachability solvers for linear ordinary differential equations  ...  In Section 4 we describe how to propagate sets using Carleman linearization. In Section 5 we extend this approach to a reachability algorithm for dissipative nonlinear dynamical systems.  ... 
arXiv:2108.10390v1 fatcat:liamuxx74zdd5l7s4biqhj2eh4

Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems

Caroline Fabre
1996 E S A I M: Control, Optimisation and Calculus of Variations  
The method of proof combines a suitable linearization of the system with a xed point argument. We then are led to study the approximate controllability of linear Stokes systems with potentials.  ...  This is done by means of Carleman estimates. (1:1) Universit e P aris 12-Val de Marne, U.F.R.  ...  In a third section, using these unique continuation properties, we will deduce the approximate controllability for linear Stokes systems and, in particular, the case of the linearized Navier-Stokes equations  ... 
doi:10.1051/cocv:1996109 fatcat:cz7b57bvrrf65fdfxrwav5f5d4

Boundary Null Controllability With Constrained Control for a Nonlinear Two Stroke System: Application to Boundary Sentinel and Identification of Parameters in a Nonlinear Population Dynamics Model

Somdouda Sawadogo, Mifiamba Soma
2019 Journal of Mathematics Research  
We first prove a new controllability result for a nonlinear two stroke system. The key to solve this controllability problem is an adapted Carleman inequality.  ...  Next, the obtained result is used to build a boundary sentinel to identify unknown parameters in a nonlinear population dynamics model with incomplete data.  ...  Using a derivation of Leray-Schauder fixed point theorem and Carleman inequality for the adjoint system, Traoré, O.  ... 
doi:10.5539/jmr.v11n4p51 fatcat:ftdlv5n3uze2vo45laibvj7n6q

Null and approximate controllability for weakly blowing up semilinear heat equations

Enrique Fernández-Cara, Enrique Zuazua
2000 Annales de l'Institut Henri Poincare. Analyse non linéar  
We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f (y) is such that |f (s)| grows slower than |s| log 3/2 (1+|s  ...  Results of the same kind are proved in the context of approximate controllability.  ...  Indeed, as usual, we first linearize the system and show its controllability analyzing how the control depends of the size of the potential of the linearized equation.  ... 
doi:10.1016/s0294-1449(00)00117-7 fatcat:phr2i5nrencojnnzpygss5nvwm

Null controllability of a nonlinear population dynamics problem

Oumar Traore
2006 International Journal of Mathematics and Mathematical Sciences  
Using a derivation of Leray-Schauder fixed point theorem and Carleman inequality for the adjoint system, we show that for all given initial density, there exists an internal control acting on a small open  ...  set of the domain and leading the population to extinction.  ...  In what follows, using a Carleman inequality for an adjoint system we establish a null controllability result for the nonlinear population dynamics models stated in (1.1) when the initial distribution  ... 
doi:10.1155/ijmms/2006/49279 fatcat:62higoe7tjh25o4w7cm7jdijd4

Behavior of the Regularized Sampling Inverse Scattering Method at Internal Resonance Frequencies

N. Shelton, K. F. Warnick
2002 Electromagnetic Waves  
NL, as it is presented in [1], has the following advantages over OLS: original nonlinear inverse problem is reduced to a finite sequence of linear ones; resulting linear problem is one-dimensional while  ...  Natural linearization (NL) was introduced in [1] as an alternative to output least-squares (OLS) approach to nonlinear parameter identification problems for partial differential equations.  ...  of interfaces using elastic waves at fixed frequency.  ... 
doi:10.2528/pier02092502 fatcat:d2xx2q47qngnzewhh6lgs66iru

The cost of approximate controllability of heat equation with dynamical boundary conditions [article]

I. Boutaayamou, S. E. Chorfi, L. Maniar, O. Oukdach
2020 arXiv   pre-print
Combining new developed Carleman estimates and some optimization techniques, we obtain explicit bounds of the minimal norm control. We consider the linear and the semilinear cases.  ...  In this paper we study the cost of approximate controllability for this equation.  ...  in L 2 and, by using the linearity and the continuity of solution operator, we get ( ϕ n , ϕ n Γ ) = Φ n ⇀ Ψ = (ψ, ψ Γ ) weakly in L 2 (0, T ; L 2 ), where Ψ is the solution of the adjoint system (4)  ... 
arXiv:2006.06711v1 fatcat:giafp6gpxnabjfgib2d7f7xt6i

Data-driven model order reduction of quadratic-bilinear systems

Ion Victor Gosea, Athanasios C. Antoulas
2018 Numerical Linear Algebra with Applications  
For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation.  ...  We illustrate the practical applicability of the proposed method by means of several numerical experiments resulting from semi-discretized nonlinear partial differential equations. † Data-Driven System  ...  a projection based moment matching MOR approach that uses the quadratic-linear representation of nonlinear systems.  ... 
doi:10.1002/nla.2200 fatcat:b2qri6ktxzaftbye6zuzdcqaxe

Internal control of the Schrödinger equation

Camille Laurent
2014 Mathematical Control and Related Fields  
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation.  ...  After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results.  ...  I also thank Ivonne Rivas and Matthieu Léautaud for useful comments about the preliminary versions of these notes.  ... 
doi:10.3934/mcrf.2014.4.161 fatcat:q4yhumgbznembp7yummy5olpn4

Internal control of the Schrödinger equation [article]

Camille Laurent
2013 arXiv   pre-print
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schr\"odinger equation.  ...  After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension 1 using some propagation results.  ...  I also thank Ivonne Rivas and Matthieu Léautaud for useful comments about the preliminary versions of these notes.  ... 
arXiv:1307.2220v1 fatcat:2mggkyjzcfg7jjyhsnqhmm2vfm

Global Controllability and Stabilization for the Nonlinear Schrödinger Equation on Some Compact Manifolds of Dimension 3

Camille Laurent
2010 SIAM Journal on Mathematical Analysis  
We prove global internal controllability in large time for the nonlinear Schrödinger equation on some compact manifolds of dimension 3.  ...  We use Bourgain spaces.  ...  We can then use this remark and the work of S. Jaffard [23] and V. Komornik [25] for the linear equation on T n to get some local nonlinear results .  ... 
doi:10.1137/090749086 fatcat:egvylrsotjaq5dro2dnd55jsxi

Insensitizing control for linear and semi-linear heat equations with partially unknown domain

Pierre Lissy, yannick privat, Yacouba Simpore
2018 E S A I M: Control, Optimisation and Calculus of Variations  
It rests upon a linearization procedure together with the use of an appropriate fixed point theorem.  ...  We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of R N (N ∈ N * ), assumed to be an unknown perturbation of a reference  ...  To conclude, let us mention that other linear or nonlinear parabolic systems coming from fluid mechanics have also been intensively studied, see for instance [4, 5, 13, 14] or [6] .  ... 
doi:10.1051/cocv/2018035 fatcat:ftopjvha4jhqtovczfwm6egzge

Approximate controllability by birth control for a nonlinear population dynamics model

Otared Kavian, Oumar Traoré
2010 E S A I M: Control, Optimisation and Calculus of Variations  
In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.  ...  In this paper we analyse an approximate controllability result for a nonlinear population dynamics model.  ...  The unique continuation result used there, is derived from a new Carleman inequality.  ... 
doi:10.1051/cocv/2010043 fatcat:qezscwndmvf7np7pzwbn2pk7ui

Boundary Stabilization of the Korteweg-De Vries Equation [chapter]

Bing-Yu Zhang
1994 Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena  
The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s.  ...  A list of open problems is also provided for further investigation.  ...  The linear result was then extended to the nonlinear system to obtain Theorem 3.1 by using the contraction mapping principle.  ... 
doi:10.1007/978-3-0348-8530-0_21 fatcat:n4jupoxc5vbdnkqwtfuptqyice

Control and stabilization of the Korteweg-de Vries equation: recent progresses

Lionel Rosier, Bing-Yu Zhang
2009 Journal of Systems Science and Complexity  
The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s.  ...  A list of open problems is also provided for further investigation.  ...  The linear result was then extended to the nonlinear system to obtain Theorem 3.1 by using the contraction mapping principle.  ... 
doi:10.1007/s11424-009-9194-2 fatcat:ww5fnubjzffvxp3jdfbao6mruq
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