A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Boundary observability and exact controllability of strongly coupled wave equations

<p style='text-indent:20px;'>In this paper, we study the exact controllability of a system of two wave equations coupled by velocities with boundary control acted on only one equation. In the first part of this paper, we consider the <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-d case. Then, using a multiplier technique, we prove that, by observing only one component of the associated homogeneous system, one can get back a full energy of both components in the case where the waves propagate with equal speeds (<i>i.e.</i> <inline-formula><tex-math id="M2">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> in (1)) and where the coupling parameter <inline-formula><tex-math id="M3">\begin{document}$ b $\end{document}</tex-math></inline-formula> is small enough. This leads, by the Hilbert Uniqueness Method, to the exact controllability of our system in any dimension space. It seems that the conditions <inline-formula><tex-math id="M4">\begin{document}$ a = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ b $\end{document}</tex-math></inline-formula> small enough are technical for the multiplier method. The natural question is then : what happens if one of the two conditions is not satisfied? This consists the aim of the second part of this paper. Indeed, we consider the exact controllability of a system of two one-dimensional wave equations coupled by velocities with a boundary control acted on only one equation. Using a spectral approach, we establish different types of observability inequalities which depend on the algebraic nature of the coupling parameter <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula> and on the arithmetic property of the wave propagation speeds <inline-formula><tex-math id="M7">\begin{document}$ a $\end{document}</tex-math></inline-formula>.</p>

Subdiffusive Source Sensing by a Regional Detection Method

Motivated by the fact that the danger may increase if the source of pollution problem remains unknown, in this paper, we study the source sensing problem for subdiffusion processes governed by time fractional diffusion systems based on a limited number of sensor measurements. For this, we first give some preliminary notions such as source, detection and regional spy sensors, etc. Secondly, we investigate the characterizations of regional strategic sensors and regional spy sensors. A regional detection approach on how to solve the source sensing problem of the considered system is then presented by using the Hilbert uniqueness method (HUM). This is to identify the unknown source only in a subregion of the whole domain, which is easier to be implemented and could save a lot of energy resources. Numerical examples are finally included to test our results.

Regional Enlarged Observability of Fractional Differential Equations with Riemann—Liouville Time Derivatives

We introduce the concept of regional enlarged observability for fractional evolution differential equations involving Riemann–Liouville derivatives. The Hilbert Uniqueness Method (HUM) is used to reconstruct the initial state between two prescribed functions, in an interested subregion of the whole domain, without the knowledge of the state.

Enlarged Controllability of Riemann–Liouville Fractional Differential Equations

We investigate exact enlarged controllability (EEC) for time fractional diffusion systems of Riemann–Liouville type. The Hilbert uniqueness method (HUM) is used to prove EEC for both cases of zone and pointwise actuators. A penalization method is given and the minimum energy control is characterized.

Controllability of a one-dimensional fractional heat equation: theoretical and numerical aspects

We analyse the controllability problem for a one-dimensional heat equation involving the fractional Laplacian $(-d_x^{\,2})^{s}$ on the interval $(-1,1)$. Using classical results and techniques, we show that, acting from an open subset $\omega \subset (-1,1)$, the problem is null-controllable for $s&gt;1/2$ and that for $s\leqslant 1/2$ we only have approximate controllability. Moreover, we deal with the numerical computation of the control employing the penalized Hilbert Uniqueness Method and a finite element scheme for the approximation of the solution to the corresponding elliptic equation. We present several experiments confirming the expected controllability properties.

Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection

We study the exact controllability of finite dimensional Galerkin approximation of a Navier-Stokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in Rd (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced undercertain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.

Ill-conditioning versus ill-posedness for the boundary controllability of the heat equation

Ill-posedness and/or ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact-controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM (Hilbert Uniqueness Method) control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations.