Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllability to the trajectories of the Navier–Stokes system.     

Localized null controllability and corresponding minimal norm control blowup rates of thermoelastic systems

In this paper, we consider the problem of null controllability for an elastic operator under square root damping. Such partial differential equation models can be described by analytic semigroups on the basic space of finite energy. Thus by inherent smoothing coming from the parabolic-like behavior of the dynamics, the problem of null controllability is appropriate for consideration. In particular, we will show that the solution variables can be steered to the zero state by means of iterations of locally supported steering controls acting on appropriate finite dimensional systems. The hinged boundary conditions considered here admit of a diagonalization of the spatial operator. The control strategy implemented in [A. Benabdallah, M. Naso, Null controllability of a thermoelastic plate, Abstr. Appl. Anal. 7 (2002) 585-599] is used to construct a suboptimal control for the problem, but here we expand upon their results by providing a bound for the energy function Emin(T), T>0. Our results are valid for localized mechanical and thermal control. The strategy relies heavily on the availability of a Carleman's estimate for finite linear combinations of eigenfunctions of the Dirichlet Laplacian.     

Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form

We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval. In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason, we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally, in a standard way, we deduce null controllability also for semilinear equations.      

Controllability of a class of heat equations with memory in one dimension

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.     

Null controllability of von Kármán thermoelastic plates under the clamped or free mechanical boundary conditions

In this paper, we provide results of local and global null controllability for 2-D thermoelastic systems, in the absence of rotational inertia, and under the influence of the (nonLipschitz) von Kármán nonlinearity. The plate component may be taken to satisfy either the clamped or higher order (and physically relevant) free boundary conditions. In the accompanying analysis, critical use is made of sharp observability estimates which obtain for the linearization of the thermoelastic plate (these being derived in [G. Avalos, I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl. 294 (2004) 34-61] and [G. Avalos, I. Lasiecka, Asymptotic rates of blowup for the minimal energy function for the null controllability of thermoelastic plates: The free case, in: Proc. of the Conference for the Control of Partial Differential Equations, Georgetown University, Dekker, in press]). Moreover, another key ingredient in our work to steer the given nonlinear dynamics is the recent result in [A. Favini, M.A. Horn, I. Lasiecka, D. Tataru, Addendum to the paper: Global existence, uniqueness and regularity of solution to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equations 10 (1997) 197-200] concerning the sharp regularity of the von Kármán nonlinearity.     

Exact null controllability of semilinear integrodifferential systems in Hilbert spaces

Sufficient conditions for exact null controllability of the semilinear integrodifferential systems in Hilbert spaces are obtained. It is shown that under some natural conditions exact null controllability of the semilinear integrodifferential system is implied by the exact null controllability of the corresponding linear system with additive term. An application to partial integrodifferential equations is given.     

Controllability of systems of Stokes equations with one control force: existence of insensitizing controls

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L² norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when the coupling terms concern first and second order derivatives, respectively), we prove this result for some general couplings.     

Exact null controllability of semilinear evolution systems

We study the exact null controllability for the abstract evolution equations in Hilbert spaces. Assuming the exact null controllability of the corresponding linearized equation we obtain sufficient conditions for the exact null controllability of the semilinear evolution equation. The results we obtained are generalization and continuation of the recent results on this issue. In the end, an example is given to show the application of our result.     

Persistent regional null controllability of some degenerate wave equations

This paper is addressed to a study of the persistent regional null controllability problems for one‐dimensional linear degenerate wave equations through a distributed controller. Different from non‐degenerate wave equations, the classical null controllability results do not hold for some degenerate wave equations. Thus, persistent regional null controllability is introduced, which means finding a control such that the corresponding state of the degenerate wave equation may vanish in a suitable subset of the space domain in a period of time. In order to solve this problem, we need to establish the regional null controllability for degenerate wave equations. This problem is reduced to a suitable observability problem of a linear degenerate wave equation. The key point is to choose a suitable multiplier in order to establish this observability inequality. Copyright © 2017 John Wiley & Sons, Ltd.     

Carleman Estimates and Controllability of Linear Stochastic Heat Equations

   Abstract. This work is concerned with Carleman inequalities and controllability properties for the following stochastic linear heat equation (with Dirichlet boundary conditions in the bounded domain D ⊂ R ^d and multiplicative noise):

Approximate controllability of Hilfer fractional differential equations

In this paper, the approximate controllability for a class of Hilfer fractional differential equations (FDEs) of order 1<α<2 and type 0 ≤ β ≤ 1 is considered. The existence and uniqueness of mild solutions for these equations are established by applying the Banach contraction principle. Further, we obtain a set of sufficient conditions for the approximate controllability of these equations. Finally, an example is presented to illustrate the obtained results.     

Controllability of the perturbed adiabatic oscillator

In this paper, we consider the global controllability with bounded controls of the perturbed adiabatic oscillator. More explicitly, we study the equationÿ+(1+g(t))y=u, whereg(t) is a suitably small perturbation term andu is the control function, which is restricted either by norm or by range. We give sufficient conditions for global null controllability, under various conditions ong(t), whenu lies in a unit ball or is constrained to assume only positive values.     

Some results on the controllability of forward stochastic heat equations with control on the drift

In this paper, we establish the null/approximate controllability for forward stochastic heat equations with control on the drift. The null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators. As a consequence of the null controllability, we obtain the observability estimate for backward stochastic heat equations, which leads to a unique continuation property for backward stochastic heat equations, and hence the desired approximate controllability for forward stochastic heat equations. It deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability. This is a new phenomenon arising in the study of the controllability problem for stochastic heat equations.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

Some controllability results for the 2D Kolmogorov equation

In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.     

The controllability problem for nonlinear Volterra systems

We discuss the controllability of systems whose dynamics are governed by a large class of nonlinear Volterra integral equations. The property of controllability is shown to be equivalent to the existence of a fixed point of a certain set-valued map. We show that convexity and seminormality conditions intimately related to those assumed in proofs of existence theorems for optimal controls are sufficient to guarantee controllability. Approximate controllability results are obtained by first introducing generalized solutions and then showing, under only mild additional restrictions, that ordinary solutions are dense in this broader class of trajectories.Dedicated to L. CesariTu se' lo mio maestro e il mio autore: Tu se' solo colui, da cui io tolsi Lo bello stile che m' ha fatto onore. Dante, Canto I: 85–87This work was written while the author was on leave to the Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, West Germany.     

Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

Exact null controllability of the Lobesia botrana model with diffusion

This paper is devoted to analyze the exact null controllability of the diffusive Lobesia botrana model with nonlocal boundary condition. We study the null controllability of butterfly population by acting on eggs, larvas and female moths in a small age interval. We assume that the control of female moths is in a certain given region of the vineyard. The main result is established by combining some estimations and the Carleman inequality for the backward system related to an optimal control problem. A fixed point theorem is then used to draw the conclusion.     

Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (Stochastic Partial Differential Equations and Applications, Series of Lecture Notes in Pure and Appl. Math., vol. 245, pp. 253–260, Chapman and Hall, London, 2006) and Goreac (Applied Analysis and Differential Equations, pp. 153–164, World Scientific, Singapore, 2007) from the finite dimensional to the infinite dimensional case.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.