Exact controllability is proven on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.
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.
The paper is devoted to the controllability problem for 3D compressible Euler system. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections
We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.
This paper is devoted to study the controllability of a one-dimensional fluid-particle interaction model where the fluid follows the viscous Burgers equation and the point mass obeys Newton's second law. We prove the null controllability for the velocity of the fluid and the particle and an approximate controllability for the position of the particle with a control variable acting only on the particle. One of the novelties of our work is the fact that we achieve this controllability result in a uniform time for all initial data and without any smallness assumptions on the initial data.
We study the controllability of a Partial Differential Equation of transport type, that arises in crowd models. We are interested in controlling it with a control being a vector field, representing a perturbation of the velocity, localized on a fixed control set. We prove that, for each initial and final configuration, one can steer approximately one to another with Lipschitz controls when the uncontrolled dynamics allows to cross the control set. We also show that the exact controllability only holds for controls with less regularity, for which one may lose uniqueness of the associated soluti…
In this paper, we study the local exact controllability to special trajectories of the micropolar fluid systems in dimension d = 2 and d = 3. We show that controllability is possible acting only on one velocity.
In this article, we study the boundary null-controllability properties of the one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$) compressible Navier-Stokes equations in the interval $(0,1)$ when a control function is acting either on the density or velocity component at one end of the interval. We first prove that the linearized system, with a Dirichlet boundary control on the density component and homogeneous Dirichlet boundary conditions on the velocity component, is null-controllable in $H^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, …
We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter sigma. More precisely the control, independent of sigma, that drives an initial data to a family of final states at time t = T, whose average in sigma is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. The resulting discrete control problem has solution whenever the continuous one has it, and we give a convergence result of the discrete controls to the c…
We study the memory-type null controllability property for wave equations involving memory terms. The goal is not only to drive the displacement and the velocity (of the considered wave) to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. This memory-type null controllability problem can be reduced to the classical null controllability property for a coupled PDE-ODE system. The later is viewed as a degenerate system of wave equations, the velocity of propagation for the ODE component vanishing. Th…
We study the controllability of linearized shape-dependent operators for flow problems. The first operator is a mapping from the shape of the computational domain to the tangential wall velocity of the potential flow problem and the second operator maps to the wall shear stress of the Stokes problem. We derive linearizations of these operators, provide their well-posedness and finally show approximate controllability. The controllability of the linearization shows in what directions the observable can be changed by applying infinitesimal shape deformations.
Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expens…
We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for a $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results.
We study the problems of controllability and ergodicity of the system of 3D primitive equations modeling large-scale oceanic and atmospheric motions. The system is driven by an additive force acting only on a finite number of Fourier modes in the temperature equation. We first show that the velocity and temperature components of the equations can be simultaneously approximately controlled to arbitrary position in the phase space. The proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the controllability of the linearisation of primitive equations around a non-s…
In this paper we study the exact controllability problem for the wave equation on a finite metric graph with the Kirchhoff-Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if $H^1(0,T)'$ Neumann controllers are placed at the active vertices and $L^2(0,T)$ Dirichlet controllers are placed at the active edges. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical …
Let $G$ be a simple, undirected graph on the vertex set $V=\{1,2,\ldots ,n\}$ and let $A$ be the adjacency matrix of $G.$ A non-empty subset $ \{i_{1},i_{2},\ldots ,i_{k}\}$ of $V$ is called a driver set for $G$ if the system $\mathbf{\dot{x}}=A\mathbf{x}+u_{1}\mathbf{e}_{i_{1}}+\cdots +u_{k} \mathbf{e}_{i_{k}}$ is controllable. In this paper we classify the minimal driver sets for the path and cycle graphs $P_{n}$ and $C_{n}$ for all values of $n$ and we determine which of those minimal driver sets render the system to be strongly structural controllable with respect to the family of all symm…
We consider the approximate control of solitons in generalized Korteweg-de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is locally null controllable, and moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight on the control of nonlinear objects, from the point of view of interaction and collision problems for no…
In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\bar ρ,0,\bar θ)$, with $\bar ρ> 0,$ $ \bar θ> 0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove tha…
Connected and automated vehicles (CAVs) have a great potential to improve traffic efficiency in mixed traffic systems, which has been demonstrated by multiple numerical simulations and field experiments. However, some fundamental properties of mixed traffic flow, including controllability and stabilizability, have not been well understood. This paper analyzes the controllability of mixed traffic systems and designs a system-level optimal control strategy. Using the Popov-Belevitch-Hautus (PBH) criterion, we prove for the first time that a ring-road mixed traffic system with one CAV and multipl…
In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbo…
In this work, we study the minimal time to steer a given crowd to a desired configuration. The control is a vector field, representing a perturbation of the crowd velocity, localized on a fixed control set. We characterize the minimal time for a discrete crowd model, both for exact and approximate controllability. This leads to an algorithm that computes the control and the minimal time. We finally present a numerical simulation.
Control of complex networks requires both structure and dynamics
School of Informatics and Computing, Indiana University（印第安纳大学信息学与计算学院）；Program in Cognitive Science, Indiana University（印第安纳大学认知科学项目）；Instituto Gulbenkian de Ciencia（古尔本基安科学研究所）
