Null Controllability Result Research Articles

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

Application of Non-Linear Evolution Stochastic Equations with Asymptotic Null Controllability Analysis

This paper investigated system of stochastic differential equations with prominence on disparities of drift parameters. These problems were solved analytical by adopting the Ito’s method of solution and three different investment solutions were obtained consequently. The necessary conditions were achieved which govern various drift parameters in assessing financial markets. Therefore, the impressions on each solution of investors in financial markets were analyzed graphically. Secondly, stock price data of Transco, LTD were analyzed which covariance matrix were considered and analysis were logically extended to stochastic vector differential equation where control measures were incorporated that would help in predicting different stock price processes, and the result obtained by exploring the properties of the fundamental matrix solution where asymptotic null controllability results were obtained by the singularity of the controllability matrix a function of the drift. Finally, the effects of the significant parameters of stochastic variables were successfully discussed.

Some controllability results for linearized compressible Navier-Stokes system with Maxwell's law

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.

Global exact controllability of the viscous and resistive MHD system in a rectangle thanks to the lateral sides and to distributed phantom forces

We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.

Controllability of a thermoelastic system

In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

This paper deals with the null-controllability of a system of mixed parabolic-elliptic pdes at any given time T>0. More precisely, we consider the Kuramoto-Sivashinsky–Korteweg-de Vries equation coupled with a second order elliptic equation posed in the interval (0,1). We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the Carleman approach, we provide the existence of a control with the explicit cost CeC/T with some constant C>0 independent in T. Then, applying the source term method developed in [39], followed by the Banach fixed point theorem, we conclude the small-time local null-controllability result of the nonlinear system.

Carleman inequality for a class of super strong degenerate parabolic operators and applications

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem.

Null controllability for a structurally damped stochastic plate equation

In this paper, we obtain a null controllability result for a structurally damped refined stochastic plate equation with two kinds of boundary conditions. The key point is to establish an observability inequality for the backward structurally damped stochastic plate equation. For this purpose, we derive a new global Carleman estimate for the damped stochastic plate operator with homogeneous Dirichlet boundary conditions on the solution and the laplacian of the solution and with homogeneous Neumann boundary conditions of first and third order, respectively. Such an estimate is obtained by using the Carleman estimate for stochastic parabolic operators with complex coefficients in an iteration manner.

Stackelberg-Nash exact controllability for the Kuramoto-Sivashinsky equation with boundary and distributed controls

This paper deals with a multi-objective control problem for the Kuramoto-Sivashinsky equation by following a Stackelberg-Nash strategy. We have a distributed control called Leader, and two boundary controls called Followers, each of them has to act over the equation to influence the behavior of the state in a particular way, by reaching or approaching to many targets at once. To be more precise, the Leader wants to drive the solution to a prescribed target at a final time, and the followers have to minimize some given cost functionals, adapting themselves to what the Leader wants. The main difficulty here is that, since the Followers are in the boundary, the problem turns to be equivalent to prove a partial null controllability result for a system of nonlinear fourth-order equations with boundary coupling terms.

Remarks on the Control of Two-Phase Stefan Free-Boundary Problems

This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions, must be satisfied; the phases are separated by a phase change interface, where an additional free-boundary condition is imposed (the so-called Stefan condition). We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove the following local null controllability result: the temperatures can be steered to zero and, simultaneously, the interface can be steered to a prescribed location provided the initial data and the interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (which gives appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (which gives the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary.

Null Controllability for a Degenerate Population Equation with Memory

In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.

"Feedback null controllability for a class of semilinear control systems"

"For a class of semilinear control systems we get null controllability results and estimates for the minimum time function by considering appropriate feedback laws."

Uncertainty principles in Gelfand-Shilov spaces and null-controllability

We provide new uncertainty principles for functions in a general class of Gelfand-Shilov spaces. These results apply, in particular, with the classical Gelfand-Shilov spaces as well as for spaces of functions with weighted Hermite expansions. Thanks to these uncertainty principles, we derive null-controllability results for evolution equations with adjoint systems enjoying smoothing effects in specific Gelfand-Shilov spaces. More precisely, we consider control subsets which are thick with respect to a quasi linearly growing density and establish sufficient conditions on the growth of the density to ensure null-controllability of these evolution equations.

The Reflection Principle in the Control Problem of the Heat Equation

We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle π/2n. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms in terms of geometric parameters of the control set.

Null controllability of coupled systems of degenerate parabolic integro-differential equations

This article concerns the null controllability of a coupled system of two degenerate parabolic integro-differential equations with one locally distributed control force. Since the memory terms do not allow applying the standards Carleman estimates directly, we start by proving a null controllability result for an associated nonhomogeneous degenerate coupled system employing new Carleman estimates with appropriate weight functions. As a consequence, we deduce the null controllability result for the initial memory system by using the Kakutani's fixed point Theorem.

Smooth Controllability of the Navier–Stokes Equation with Navier Conditions: Application to Lagrangian Controllability

We deal with the 3D Navier–Stokes equation in a smooth simply connected bounded domain, with controls on a non-empty open part of the boundary and a Navier slip-with-friction boundary condition on the remaining, uncontrolled, part of the boundary. We extend the small-time global exact null controllability result in Coron et al. (J Eur Math Soc 22:1625–1673, 2020) from Leray weak solutions to the case of smooth solutions. Our strategy relies on a refinement of the method of well-prepared dissipation of the viscous boundary layers which appear near the uncontrolled part of the boundary, which allows to handle the multi-scale features in a finer topology. As a byproduct of our analysis we also obtain a small-time global approximate Lagrangian controllability result, extending to the case of the Navier–Stokes equations the recent results (Glass and Horsin in J Math Pures Appl (9) 93:61–90, 2010; Glass and Horsin in SIAM J Control Optim 50: 2726–2742, 2012; Horsin and Kavian in ESAIM Control Optim Calc Var 23:1179–1200, 2017) in the case of the Euler equations and the result (Glass and Horsin in ESAIM Control Optim Calc Var 22:1040–1053, 2016) in the case of the steady Stokes equations.

Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions

<p style='text-indent:20px;'>This paper deals with the hierarchical control of the anisotropic heat equation with dynamic boundary conditions and drift terms. We use the Stackelberg-Nash strategy with one leader and two followers. To each fixed leader, we find a Nash equilibrium corresponding to a bi-objective optimal control problem for the followers. Then, by some new Carleman estimates, we prove a null controllability result.</p>

Null controllability for a class of stochastic singular parabolic equations with the convection term

<p style='text-indent:20px;'>This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.</p>

Controllability for degenerate/singular parabolic systems involving memory terms

<p style='text-indent:20px;'>In this paper we deal with the null controllability for degenerate/singular parabolic systems with memory terms. To this aim, we first prove the null controllability property for some auxiliary nonhomogeneous degenerate/singular problems via new Carleman estimates for their corresponding adjoint systems. Then, under a condition on the kernels, using the Kakutani's fixed point theorem, we deduce null controllability results for the initial problems with memory.</p>

Null Controllability of the Parabolic Spherical Grushin Equation

We investigate the null controllability property of the parabolic equation associated with the Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere 𝕊2. This is the natural generalization of the Grushin operator 𝒢 = ∂x2 + x2∂y2 on ℝ2 to this curved setting and presents a degeneracy at the equator of 𝕊2. We prove that the null controllability is verified in large time when the control acts as a source term distributed on a subset ω̅ = {(x1, x2, x3) ∈ 𝕊2 | α < | x3 | < β} for some 0 ≤ α < β ≤ 1. More precisely, we show the existence of a positive time T* > 0 such that the system is null controllable from ω̅ in any time T ≥ T*, and that the minimal time of control from ω̅ satisfies Tmin ≥ log(1/√(1 - α2)) . Here, the lower bound corresponds to the Agmon distance of ω̅ from the equator. These results are obtained by proving a suitable Carleman estimate using unitary transformations and Hardy-Poincaré type inequalities to show the positive null-controllability result. The negative statement is proved by exploiting an appropriate family of spherical harmonics, concentrating at the equator, to falsify the uniform observability inequality.