Exact Controllability for Wave Equation on General Metric Graphs with Non-smooth Controls

In this paper, we consider initial boundary value problems and control problems for the wave equation on finite metric graphs with Dirichlet boundary controls. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the full controllability utilizes both dynamical and moment method approaches.

In the present paper, we are concerned with a fractional wave equation of Sturm–Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler–Lagrange first-order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first-order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of \(2\)-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger \(2\)-coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).

Exact Controllability for the Time Dependent Transport Equation

All articles must In this paper we introduce some new concepts for second-order hyperbolic equations: two-point boundary value problem, global exact controllability and exact controllability. For several kinds of important linear and nonlinear wave equations arising from physics and geometry, we prove the existence of smooth solutions of the two-point boundary value problems and show the global exact controllability of these wave equations. In particular, we investigate the two-point boundary value problem for one-dimensional wave equation defined on a closed curve and prove the existence of smooth solution which implies the exact controllability of this kind of wave equation. Furthermore, based on this, we study the two-point boundary value problems for the wave equation defined on a strip with Dirichlet or Neumann boundary conditions and show that the equation still possesses the exact controllability in these cases. Finally, as an application, we introduce the hyperbolic curvature flow and obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves.

When the rotatory inertia is taken into account, vibrations of a linear plate can be described by the Kirchhoff plate equation. Consider this equation with locally distributed control forces and some boundary condition which is the simply supported boundary condition for a rectangular plate. In this paper, the authors establish exact controllability of the system in terms of the equivalence to exact internal controllability of the wave equation, by means of a frequency domain characterization of exact controllability introduced recently in [11].

To describe the flow of a miscible quantity on a network, we consider the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. The structure of the graph influences strongly the dynamics. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations using a basis of eigenvectors of the graph Laplacian. These lead us to introduce the notion of soft nodes. We give sufficient conditions for their existence in general graphs. They can cause several effects as we show on small graphs, for example the ineffectiveness of damping applied to them. Soft nodes may be of critical importance for complex physical networks and engineering networks like power grids.

The aim of this paper is to prove new exact controllability properties of systems described by perturbations of the classical Kirchhoff plate equation. We first consider systems described by an abstract plate equation with a bounded control operator. The generator of these systems is perturbed by bounded operators which are not necessarily compact, thus not falling in the range of application of compactness-uniqueness arguments. Our first main result is abstract and can be informally stated as follows: if the system described by the corresponding unperturbed abstract wave equation, with the same control operator, is exactly controllable (in some time), then the considered perturbed plate system is exactly controllable in arbitrarily small time. The employed methodology is based, in particular, on frequency-dependent Hautus type tests for systems with skew-adjoint operators. When applied to systems described by the classical Kirchhoff equations, our abstract results, combined with some elliptic Carleman-type estimates, yield exact controllability in arbitrarily small time, provided that the system described by the wave equation in the same spatial domain and with the same control operator is exactly controllable. The same abstract results can be used to prove the exact controllability of the system obtained by linearizing the von Kármán plate equation around a real analytic stationary state. This leads, via a fixed-point method, to our second main result: the nonlinear system described by the von Kármán plate equations is locally exactly controllable around any stationary state defined by a real analytic function. We also discuss the possible application of the methods in this paper to systems described by Schrödinger type equations on manifolds or by the related Berger’s nonlinear plate equation.

ABSTRACTIn this paper, we study the robust finite-time containment control problem for a class of high-order uncertain nonlinear multi-agent systems modelled as high-order integrator systems with bounded matched uncertainties. When relative state information between neighbouring agents is available, an observer-based distributed controller is proposed for each follower using the sliding mode control technique which solves the finite-time containment control problem under general directed communication graphs. When only relative output information is available, robust exact differentiators and high-order sliding-mode controllers are employed together with the distributed finite-time observers. It is shown that robust finite-time containment control can still be achieved in this situation. An application in the coordination of multiple non-holonomic mobile robots is used as an example to illustrate the effectiveness of the proposed control strategies.

Avdonin and Kurasov proposed a leaf peeling method based on the boundary control to recover a potential for the wave equation on a tree. Avdonin and Nicaise considered a source identification problem for the wave equation on a tree. This paper extends the methodology to the wave equation with unknown potential and source distributed parameters defined on a general tree graph.

In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step $h\rightarrow 0$, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to $h\rightarrow 0$) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.

In this paper, by using the Hilbert Uniqueness Method (HUM), we study the exact controllability problem described by the wave equation in a three-dimensional horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities whose size depends on a small parameter ε>0, and with a fixed height. Our aim is to obtain the exact controllability for the homogenized equation. In the process, we study the asymptotic analysis of wave equation in two setups, namely solution by standard weak formulation and solution by transposition method.

n this paper we summarize our recent results on the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. It is shown that the projection of the solution in an appropriate space in which the high frequencies have been filtered is exactly controllable with uniformly bounded controls (with respect to the time-step). By classical duality arguments, the problem is reduced to a boundary observability inequality for a time-discrete wave equation. Using multiplier techniques the uniform observability property is proved in a class of filtered initial data. The optimality of the filtering parameter is also analyzed.

Exact controllability for semilinear wave equations in one space dimension

Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.