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 article is devoted to study the interior approximated controllability of the strongly damped semilinear wave equation with memory, impulses and delay terms. The problem is challenging since the state equation contains memory and impulsive terms yielding to potential unbounded control sequences steering the system to a neighborhood of the final state, thus fixed point theorems cannot be used directly. As alternative, the A.E Bashirov and et al. techniques are applied and together with the delay allow the control solution to be directed to fixed curve in a short time interval and achieve ou…
In this paper, we study structural controllability of a linear time invariant (LTI) composite system consisting of several subsystems. We assume that the neighbourhood of each subsystem is unconstrained, i.e., any subsystem can interact with any other subsystem. The interaction links between subsystems are referred as interconnections. We assume the composite system to be structurally controllable if all possible interconnections are present, and our objective is to identify the minimum set of interconnections required to keep the system structurally controllable. We consider structurally iden…
We study global controllability of 'rotating' multidimensional rigid body (MRB) controlled by application of few torques. Study by methods of geometric control requires analysis of algebraic structure introduced by the quadratic term of Euler-Frahm equation. We discuss problems, which arise in the course of this analysis, and establish several global controllability criteria for damped and non damped cases.
We consider the observability problem for non-autonomous evolution systems (i.e., the operators governing the system depend on time). We introduce an averaged Hautus condition and prove that for skew-adjoint operators it characterizes exact observability. Next, we extend this to more general class of operators under a growth condition on the associated evolution family. We give an application to the Schrödinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient.
We study the internal controllability of the semilinear wave equation $$v_{tt}(x,t)-Δv(x,t) + f(x,v(x,t))= \Un_ω u(x,t)$$ for some nonlinearities $f$ which can produce several non-trivial steady states. One of the usual hypotheses to get global controllability, is to assume that $f(x,v)v\geq 0$. In this case, a stabilisation term $u=γ(x)v_t$ makes any solution converging to zero. The global controllability then follows from a theorem of local controllability and the time reversibility of the equation. In this paper, the nonlinearity $f$ can be more general, so that the solutions of the damped …
In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix $L^*$ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs, while it is always simple for the Laplacian $L$ with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with $L$. For…
In the paper, we introduce and address the problem of structural averaged controllability for linear ensemble systems. We provide examples highlighting the differences between this problem and others. In particular, we show that structural averaged controllability is strictly weaker than structural controllability for single (or ensembles of) linear systems. We establish a set of necessary or sufficient conditions for sparsity patterns to be structurally averaged controllable.
This paper studies structural controllability for a networked dynamic system (NDS), in which each subsystem may have different dynamics, and unknown parameters may exist both in subsystem dynamics and in subsystem interconnections. In addition, subsystem parameters are parameterized by a linear fractional transformation (LFT). It is proven that controllability keeps to be a generic property for this kind of NDSs. Some necessary and sufficient conditions are then established respectively for them to be structurally controllable, to have a fixed uncontrollable mode, and to have a parameter depen…
This paper deals with structural controllability of leader-follower networks. The system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero, while the remaining entries take nonzero values. The network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix, the system is controllable in the classical sense. In this paper we introduce a more general notion of strong structural controllability which deals with the situation that given nonzero entries in the system's patte…
We consider minimal controllability problems (MCPs) on linear structural descriptor systems. We address two problems of determining the minimum number of input nodes such that a descriptor system is structurally controllable. We show that MCP0 for structural descriptor systems can be solved in polynomial time. This is the same as the existing results on typical structural linear time invariant (LTI) systems. However, the derivation of the result is considerably different because the derivation technique of the existing result cannot be used for descriptor systems. Instead, we use the Dulmage--…
This paper addresses questions regarding controllability for `generic parameter' dynamical systems, i.e. the question whether a dynamical system is `structurally controllable'. Unlike conventional methods that deal with structural controllability, our approach uses an undirected graph: the behavioral approach to modelling dynamical systems allows this. Given a system of linear, constant coefficient, ordinary differential equations of any order, we formulate necessary and sufficient conditions for controllability in terms of weights of the edges in a suitable bipartite graph constructed from % …
We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite…
This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maxim…
Recent years have witnessed a wave of research activities in systems science toward the study of population systems. The driving force behind this shift was geared by numerous emerging and ever-changing technologies in life and physical sciences and engineering, from neuroscience, biology, and quantum physics to robotics, where many control-enabled applications involve manipulating a large ensemble of structurally identical dynamic units, or agents. Analyzing fundamental properties of ensemble control systems in turn plays a foundational and critical role in enabling and, further, advancing th…
We study the strong structural controllability (SSC) of diffusively coupled networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace, which is equal to the smallest possible rank of controllability matrix under admissible (positive) coupling weights. In this paper, we compare two tight lower bounds on the dimension of strong structurally controllable subspace: one based on the distances of followers to leaders, and the other based on the graph c…
We study a damped scalar conservation law driven by the sum of a fixed external force and a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. It is proved that any solution of the uncontrolled equation can be exponentially stabilised. As a consequence, we obtain the global approximate controllability to trajectories by a one-dimensional localised control.
Problems involving control of large ensmebles of structurally identical dynamical systems, called \emph{ensemble control}, arise in numerous scientific areas from quantum control and robotics to brain medicine. In many of such applications, control can only be implemented at the population level, i.e., through broadcasting an input signal to all the systems in the population, and this new control paradigm challenges the classical systems theory. In recent years, considerable efforts have been made to investigate controllability properties of ensemble systems, and most works emphasized on linea…
Actuator placement is a fundamental problem in control design for large-scale networks. In this paper, we study the problem of finding a set of actuator positions by minimizing a given metric, while satisfying a structural controllability requirement and a constraint on the number of actuators. We first extend the classical forward greedy algorithm for applications to graphs that are not necessarily strongly connected. We then improve this greedy algorithm by extending its horizon. This is done by evaluating the actuator position set expansions at the further steps of the classical greedy algo…
In this article, a new notion of modal strong structural controllability is introduced and examined for a family of LTI networks. These networks include structured LTI subsystems, whose system matrices have the same zero/nonzero/arbitrary pattern. An eigenvalue associated with a system matrix is controllable if it can be directly influenced by the control inputs. We consider an arbitrary set Δ, and we refer to a network as modal strongly structurally controllable with respect to Δif, for all systems in a specific family of LTI networks, every λ\inΔis a controllable eigenvalue. For this family …
We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $μ_a>0$. We establish observability inequalities for weakly (when $μ_a \in [0,1[$) as well as strongly (when $μ_a \in [1,2[$) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when $μ_a>2$) and the blow-up of the observability time when $μ_a$ converges to $2$ from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degene…
A Framework for Structural Input/Output and Control Configuration Selection in Large-Scale Systems
Carnegie Mellon University（卡内基梅隆大学）；Institute for System and Robotics, Instituto Superior Técnico, Technical University of Lisbon（里斯本理工大学高等技术学院系统研究所）；Department of Electrical and Computer Engineering, Faculty of Engineering, University of Porto（波尔图大学工程学院电气与计算机工程系）
