A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions

In this paper we present a local result on the existence of insensitizing controls for a semilinear heat equation when nonlinear boundary conditions of the form ∂ny + f(y)=0 are considered. The problem leads to an analysis of a special type of nonlinear null controllability problem. A sharp study of the linear case and a later application of an appropriate fixed point argument constitute the scheme of the proof of the main result. The boundary conditions we are dealing with lead us to seek a fixed point, and thus also control functions, in certain H¨older spaces. The main strategy in this paper is the construction of controls with H¨olderian regularity starting from L2-controls in the linear case. Sufficient regularity in the data and appropriate assumptions on the right-hand side term ξ of the equation are required.Ministerio de Educación y Cienci

Global sensitivity analysis for the boundary control of an open channel

The goal of this paper is to solve the global sensitivity analysis for a particular control problem. More precisely, the boundary control problem of an open-water channel is considered, where the boundary conditions are defined by the position of a down stream overflow gate and an upper stream underflow gate. The dynamics of the water depth and of the water velocity are described by the Shallow Water equations, taking into account the bottom and friction slopes. Since some physical parameters are unknown, a stabilizing boundary control is first computed for their nominal values, and then a sensitivity anal-ysis is performed to measure the impact of the uncertainty in the parameters on a given to-be-controlled output. The unknown physical parameters are de-scribed by some probability distribution functions. Numerical simulations are performed to measure the first-order and total sensitivity indices

Insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient

In this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of IRN . We first consider a semilinear heat equation involving gradient terms with homogeneous Dirichlet boundary conditions. Then a heat equation with a nonlinear term F(y) and linear boundary conditions of Fourier type is considered. The nonlinearities are assumed to be globally Lipschitz-continuous. In both cases, we prove the existence of controls insensitizing the L2−norm of the observation of the solution in an open subset O of the domain, under suitable assumptions on the data. Each problem boils down to a special type of null controllability problem. General observability inequalities are proved for linear systems similar to the linearized problem. The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.Ministerio de Ciencia y Tecnologí

Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity

In this paper we consider a semilinear heat equation (in a bounded domain Ω of IRN ) with a nonlinearity that has a superlinear growth at infinity. We prove the existence of a control, with support in an open set ω ⊂ Ω, that insensitizes the L2−norm of the observation of the solution in another open subset O ⊂ Ω when ω ∩ O 6= ∅, under suitable assumptions on the nonlinear term f(y) and the right hand side term ξ of the equation. The proof, involving global Carleman estimates and regularizing properties of the heat equation, relies on the sharp study of a similar linearized problem and an appropriate fixed-point argument. For certain superlinear nonlinearities, we also prove an insensitivity result of a negative nature. The crucial point in this paper is the technique of construction of L r–controls (r large enough) starting from insensitizing controls in L 2.Ministerio de Educación y Cienci

Identification of the class of initial data for the insensitizing control of the heat equation

This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see[19] and[1], respectively). But in the context of pure insensitization there are very few results identifying the class of initial data that can be insensitized. This is a delicate issue which is related to the fact that insensitization turns out to be equivalent to suitable observability estimates for a coupled system of heat equations, one being forward and the other one backward in time. The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions, which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the system, one of them being an obstruction at the initial time t = 0 and the other one at the final one t = T. In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in a class that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in[14] where it is shown that smooth and compactly supported initial data may not be controlled. Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions

Insensitizing controls for a fourth order semi-linear parabolic equations

This paper is concerned with the existence of insensitizing controls for a fourth order semilinear parabolic equation. Here, the initial data is partially unknown, we would like to find controls such that a specific functional is insensitive for small perturbations of the initial data. In general, this kind of problems can be recast as a null controllability problem for a nonlinear cascade system. We will first prove a null controllability result for a linear problem by global Carleman estimates and dual arguments. Then, by virtue of Leray-Schauder's fixed points theorem, we conclude the null controllability for the cascade system in the semi-linear case.Comment: arXiv admin note: text overlap with arXiv:2211.00428, arXiv:2211.0064

Controls insensitizing the norm of solution of a Schr\"odinger type system with mixed dispersion

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schr\"odinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schr\"odinger system of cascade type with zero order coupling which is equivalent to the insensitizing control problem. Precisely, by means of new Carleman estimates, we first obtain a null controllability result for the linearized system around zero, then the null controllability for the nonlinear case is extended using an inverse mapping theorem.Comment: 26 pages. Comments are welcom

Insensitizing controls for the Cahn-Hilliard type equation

This paper is addressed to showing the existence of insensitizing controls for the one-dimensional Cahn--Hilliard type equation with a superlinear nonlinearity. We solve this problem by reducing the original problem to a controllability problem. The crucial point in this paper is an observability estimate for a linearized cascade system of the Cahn--Hilliard type equation. In order to obtain this observability estimate, we establish a global Carleman estimate for a fourth order parabolic operator