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ABSTRACT This paper concerns the null controllability of stochastic degenerate parabolic equation with convection term in weakly degenerate case. Due to the degeneracy, we first transfer to study an approximate nondegenerate system. Next, we establish the Carleman estimate for backward stochastic degenerate parabolic equation with convection term. In order to deal with the convection term, the Carleman estimate is established by introducing an auxiliary function to transfer the diffusion and the convection term into a whole complex divergence‐like form. By this Carleman estimate, we then obtain an observability inequality for the adjoint system of the approximate system. Based on the observability inequality and an approximate argument, we proved our null controllability result. [ABSTRACT FROM AUTHOR]

The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: $ \small {\sum _{j=1}^m}\frac {\mu _j}{|x-x^j|^2}, $ ∑ j = 1 m μ j | x − x j | 2 , where $ x^1, x^2,\ldots,x^m $ x 1 , x 2 , ... , x m are different points, defined in a bounded $ C^2 $ C 2 domain $ \Omega \subset \mathbb R^n $ Ω ⊂ R n , and $ \mu _1,\ldots,\mu _m $ μ 1 , ... , μ m are real constants. More precisely, we show that for any $ \mu _j\leq \frac {(n-2)^2}{4},\ j=1,\ldots,m, $ μ j ≤ (n − 2) 2 4 , j = 1 , ... , m , we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is to establish the internal null controllability for the one-dimensional heat equation featuring dynamic boundary conditions. This achievement is realized by introducing a new Carleman estimate and an observability inequality for the corresponding backward system. In conclusion, the paper includes a set of numerical experiments that serve to confirm the validity of the theoretical findings and underscore the effectiveness of the designed control with a minimal |$L^{2}$| -norm. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability of systems of $ n $-coupled parabolic equations with dynamic boundary conditions, where the coupling and control matrices $ A $ and $ B $ are constant in time and space. Being different to the case of static boundary conditions, we will show that the Kalman rank condition $ rank[B, AB,\dots, A^{n-1}B ] = n $ is a sufficient condition, we also show that it is necessary for the null controlability under an extra assumption on the boundary coupling. The null controlability result will be proved by proving Carleman and observability inequalities for the corresponding adjoint problem. [ABSTRACT FROM AUTHOR]

We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the "gene type" of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed by one control force. To reach our goal, we develop first a Carleman type inequality for its adjoint system and consequently the pertinent observability inequality. Note that such a system is obtained via the original paradigm using the Lagrangian method. Afterwards, with the help of a cost function we will be able to deduce the existence of a control acting on a subset of the gene type domain and which steers both populations of a certain class of age to extinction in a finite time.\looseness-2

This paper concerns the null controllability of an one-dimensional degenerate parabolic equation with a slightly superlinear reaction term in a bounded interval, with a control on a subinterval. The degeneracy of the equation is relatively weak, and the reaction term grows relatively slow at infinity. It is noted that the behavior of the reaction term at infinity is consistent with the well-known null controllability results for heat equations. The Carleman estimate and the observability inequality for the corresponding linear conjugate equations are established. Subsequently, the control problem is proved to be null controllable by using the observability inequality, the the Kakutani fixed point theorem and some a priori estimates. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying two multiobjective problems for stochastic degenerate parabolic equations. The first one is a hierarchical control problem, in which the controls are classified into a pair of leaders and a pair of followers. For each pair of leaders, a Nash equilibrium is searched for a noncooperative game problem. The aim of the pair of leaders is to achieve null controllability of the system. The other multiobjective problem is an inverse initial problem under a Nash equilibrium strategy. In contrast to the classical inverse initial problem, an optimization problem for stochastic degenerate parabolic equations is first investigated. Then, the conditional stability of determining initial information is derived through terminal observation. [ABSTRACT FROM AUTHOR]

We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the “gene type” of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed by one control force. To reach our goal, we develop first a Carleman type inequality for its adjoint system and consequently the pertinent observability inequality. Note that such a system is obtained via the original paradigm using the Lagrangian method. Afterwards, with the help of a cost function we will be able to deduce the existence of a control acting on a subset of the gene type domain and which steers both populations of a certain class of age to extinction in a finite time. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the boundary controllability of heat equation with dynamic boundary conditions. More precisely, we prove that the equation is null controllable at any positive time by means of a boundary control supported on an arbitrary subboundary. The proof of the main result combines a new boundary Carleman estimate and some regularity estimates for the adjoint system, with an explicit dependence with respect to the final time. This technique allows us to overcome a new difficulty that arises when absorbing a normal derivative term. [ABSTRACT FROM AUTHOR]

We establish the null controllability for fourth order stochastic parabolic equations. Utilizing the duality argument, the null controllability is reduced to the observability for fourth order backward stochastic parabolic equations, and the desired observability estimate is obtained by a new global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a fourth order stochastic parabolic operator. [ABSTRACT FROM AUTHOR]

The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: ∂ t u (x , t) − d i v (p (x) ∇ u (x , t)) − (μ / | x | 2) u (x , t) = f (x , t). Here μ is a real constant. It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when 0 ≤ μ ≤ p 1 (n − 2) 2 / 4 , and in this paper, we mainly consider the case 0 ≤ μ < (p 1 2 / p 2) (n − 2) 2 / 4 , where p 1 , p 2 are two positive constants which satisfy: 0 < p 1 ≤ p (x) ≤ p 2 , ∀ x ∈ Ω ¯ . We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996–2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317] to the system. We obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case μ > p 2 (n − 2) 2 / 4. We consider a sequence of regularized potentials μ / (| x | 2 + ϵ 2) , and prove that we cannot stabilize the corresponding systems uniformly with respect to ϵ > 0.. [ABSTRACT FROM AUTHOR]

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ > 0, when our control domain touches the degeneracy line {x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem. [ABSTRACT FROM AUTHOR]

This article concerns the null controllability of a nonlinear control system governed by coupled degenerate parabolic equations. We first establish the Carleman estimate and the observability inequality for solutions to the conjugate problem. Then we can prove the observability inequality and the null controllability of the linear control system. Finally, the nonlinear control system is shown to be null controllable by a fixed point argument and compact estimates.

This article concerns the null controllability of a control system governed by coupled degenerate parabolic equations with lower order terms. For these equations, the convection terms cannot be controlled by the diffusion terms. We establish a Carleman estimate and an observability inequality by using Carleman estimates for single degenerate parabolic equations with lower order term and some energy estimates. Then we prove that the system with two controls is null controllable. Finally, we show the null controllability of the system with one control, by constructing suitable controls.

This article shows Carleman estimate and null controllability of a cascade control system governed by the semilinear degenerate parabolic equations with the general convection terms. The semilinear parabolic equations are weakly degenerate on the boundary and the convection terms cannot be controlled by the diffusion terms. We establish the Carleman estimate and the observability inequality for the linear conjugate system, and prove that the control system is null controllable.

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations.

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called Kalman's rank condition. [ABSTRACT FROM AUTHOR]

We consider the heat equation in a bounded domain of with distributed control (supported on a small open subset) subject to dynamic boundary conditions of surface diffusion type and involving drift terms on the bulk and on the boundary. We prove that the system is null controllable at any time. The result is based on new Carleman estimates for this type of boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time.

This paper concerns the null controllability of a system of m linear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subset ω of Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Kawahara equation posed on a bounded interval with a distributed control. First, we establish a Carleman estimate for the Kawahara equation with internal observation. Then, applying this Carleman estimate, we can show that the Kawahara equation is null controllable. Furthermore, we prove the null controllability with constraints on the state for the Kawahara equation. [ABSTRACT FROM AUTHOR]

This paper concerns a class of control systems governed by degenerate parabolic equations with the gradient terms, which are independent of the diffusion terms. The Carleman estimates and the observability inequalities for the equations are established when the degeneracy is relatively weak. Subsequently, it is proved that the control systems are null controllable. Moreover, the result can be generalized to the semilinear equations by using the fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper we will generalize the Kalman rank condition for the null controllability to n-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and m-controls. For that we prove a global Carleman estimate for the solutions of a scalar 2n-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability. [ABSTRACT FROM AUTHOR]

In this paper, we consider control systems governed by a class of semilinear parabolic equations, which are singular at the boundary and possess singular convection and reaction terms. The systems are shown to be null controllable by establishing Carleman estimates, observability inequalities and energy estimates for solutions to linearized equations. [ABSTRACT FROM AUTHOR]

This paper concerns a control system governed by a convection-diffusion equation, which is weakly degenerate at the boundary. In the governing equation, the convection is independent of the degeneracy of the equation and cannot be controlled by the diffusion. The Carleman estimate is established by means of a suitable transformation, by which the diffusion and the convection are transformed into a complex union, and complicated and detailed computations. Then the observability inequality is proved and the control system is shown to be null controllable. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new global Carleman estimates for the linear stochastic Kuramoto–Sivashinsky equations. The first one is for the backward linear stochastic Kuramoto–Sivashinsky equation. Based on this estimate and the duality argument, we obtain the null controllability of the forward linear stochastic Kuramoto–Sivashinsky equation by three controls, one is an internal control in the diffusion term and the others are boundary controls. The second one is for the forward linear stochastic Kuramoto–Sivashinsky equation. According to this estimate, we obtain a unique continuation property for the forward linear stochastic Kuramoto–Sivashinsky equation. [ABSTRACT FROM AUTHOR]

In this article, we study the null controllability of a linear model with degenerate diffusion in population dynamics. We develop first a Carleman type inequality for the adjoint system of an intermediate model, and then an observability inequality. By a fixed point technique, we establish the existence of a control acting on a subset of the space domain that leads the population of a certain age to extinction in a finite time.

This paper deals with the null controllability of the system governed by coupled degenerate equations with the first order terms. The null controllability of the system with two controls is proved firstly, then the system with one control is shown to be null controllable by constructing the control function. To investigate the null controllability, we establish the Carleman estimate and the observability inequality for the adjoin system by using the Carleman estimate for the single degenerate equation. [ABSTRACT FROM AUTHOR]

In this paper we will analyze the null controllability properties of a linear coupled degenerate parabolic system of n equations when m distributed controls are exerted on the system. First we start with the case when the coupling matrix A is cascade, and then when A is a full matrix, we will prove that the Kalman rank condition on the coupling and the control matrices A and B characterizes the controllability properties of the system. [ABSTRACT FROM AUTHOR]

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time. [ABSTRACT FROM AUTHOR]

This paper is concerned with the control properties of the Korteweg-de Vries (KdV) equation posed on a bounded interval (0, L) with a distributed control. When the control region is an arbitrary open subdomain (l1, l2), we prove the null controllability of the KdV equation by means of a new Carleman inequality. As a consequence, we obtain a regional controllability result, which roughly tells us that any target function arbitrarily chosen on (0, l1) and null on (l2, L) is reachable. Finally, when the control region is a neighborhood of the right endpoint, an exact controllability result in a weighted L2-space is also established. [ABSTRACT FROM AUTHOR]

We study the large time behavior of mild solutions of HJB equations in infinite dimension by a purely probabilistic approach. For that purpose, we show that the solution of a backward SDE in finite horizon T taken at initial time behaves like a linear term in T shifted with the solution of the associated ergodic backward SDE taken at initial time. Moreover we give an explicit rate of convergence, which seems to be new to our best knowledge. [ABSTRACT FROM AUTHOR]

In this paper we discuss the observability estimates and the null controllability of forward and backward linear stochastic Kuramoto-Sivashinsky equations with random noise. The key point is to establish the Carleman estimates for stochastic fourth order forward and backward parabolic equations. [ABSTRACT FROM AUTHOR]

This paper concerns a class of control systems governed by semilinear degenerate parabolic equations with convection terms, which cannot be controlled by the diffusion terms. The Carleman estimates and the observability inequalities for the corresponding linear equations are established if the degeneracy is relatively weak. Subsequently, it is proved that the control systems are null controllable. [ABSTRACT FROM AUTHOR]

We study the null controllability properties of some degenerate/singular parabolic equations in a bounded interval of ℝ. For this reason we derive a new Carleman estimate whose proof is based on Hardy inequalities. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns the null controllability of the system governed by coupled degenerate equations. By the Carleman estimate for the case of a single degenerate equation, the Carleman estimate and the observability inequality are established. Then, the system with two controls and the system with one control are shown to be null controllable. [Copyright &y& Elsevier]

Abstract: The paper is devoted to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators (with real functions α and β), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg–Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimates. [Copyright &y& Elsevier]

Abstract: The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators. [Copyright &y& Elsevier]

In this article we consider a control problem of a linear Euler-Bernoulli damped beam equation with potential in dimension one with periodic boundary conditions. We derive a new Carleman estimate for an adjoint of the equation under consideration. Then using a well known duality argument we obtain explicitly the control function which can be used to drive the solution trajectory of the control problem to zero state. [ABSTRACT FROM AUTHOR]