In this paper we establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and observability are obtained by a constructive method. Moreover, with the linearized Saint-Venant system and the 1-D linear wave equation as examples, we show that the number of both boundary controls and boundary observations can not be reduced, and consequently, we conclude that the exact boundary controllability for a hyperbolic system in a network with loop can not be realized generically.
By means of a direct and constructive method based on the theory of semiglobal $C^2$ solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.
The general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows in certain situations to recover non zero characteristic speeds.
Controllability and observability are important properties of a distributed paramater systems.The equivalence between the notion of exact observability and exact controllability holds in general. In this work, we define a new notion of controllability say weak which is related to some weak observability inequality and we give the equivalence between.
In this Note, we review the main existing results, methods, and some key open problems on the controllability of nonlinear hyperbolic and parabolic equations. Especially, we describe our recent universal approach to solve the local controllability problem of quasilinear time-reversible evolution equations, which is based on a new unbounded perturbation technique. It is also worthy to mention that the technique we develop can also be applied to other problems for quasilinear equations, say local existence, stabilization, etc.
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.
The controllability and observability of Boolean control network(BCN) are two fundamental properties. But the verification of latter is much harder than the former. This paper considers the observability of BCN via controllability. First, the set controllability is proposed, and the necessary and sufficient condition is obtained. Then a technique is developed to convert the observability into an equivalent set controllability problem. Using the result for set controllability, the necessary and sufficient condition is also obtained for the observability of BCN.
Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of $α$-null controllability. We comment on the relationships between those various concepts, at the light of the observability inequalities that characterize…
We consider exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system, and then applying the Leray-Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so called global controllability).
This paper is concerned with the notions of admissibility, exact controllability, exact observability and regularity of linear systems in the Banach space setting. It is proved that admissible controllability, exact controllability, admissible observation, exact observability and regularity are invariant under some regular perturbations of the generators, such results are generalizations of some previous references. Moreover, the related boundary linear systems and some illustrative examples are presented.
This paper is addressed to studying the exact controllability for stochastic transport equations by two controls: one is a boundary control imposed on the drift term and the other is an internal control imposed on the diffusion term. By means of the duality argument, this controllability problem can be reduced to an observability problem for backward stochastic transport equations, and the desired observability estimate is obtained by a new global Carleman estimate. Also, we present some results about the lack of exact controllability, which show that the action of two controls is necessary. T…
The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equation…
Poset-causal systems form a class of decentralized systems introduced by Shah and Parrilo [32] and studied mainly in the context of optimal decentralized control. In this paper we develop part of the realization theory for poset-causal systems. More specifically, we investigate several notions of controllability and observability, and their relation under duality. These new notions extend concepts of controllability and observability in the context of coordinated linear systems [16]. While for coordinated linear systems there is a clear hierarchical structure with a single (main) coordinator, …
In this paper we investigate the controllability and observability properties of a family of linear dynamical systems, whose structure is induced by the Laplacian of a grid graph. This analysis is motivated by several applications in network control and estimation, quantum computation and discretization of partial differential equations. Specifically, we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph. For each eigenvalue, based on its multiplicity and on suitable symmetries of the corresponding eigenvectors, we provide necessary and suffici…
In this paper, we establish some important results for the impulsive wave equation. We begin by proving the existence of a solution. Then, we study the impulse approximate controllability where the control function acts on a subdomain $ω$ and at one instant of time $τ\in (0, T)$. Afterward, we study impulse observability
We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems of the form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) = x_0\in L_p (\mathbb{R}^d)$, with interior control on a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, and where $A$ is an elliptic operator of order $m \in \mathbb{N}$ in $L_p(\mathbb{R}^d)$. We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty …
The problem of exact observability is analyzed for a wide class of neutral type systems by an infinite dimensional approach. The duality with the exact controllabil-ity problem is the main tool. It is based on an explicit expression of a neutral type system which corresponding to the abstract adjoint system. A nontrivial relation is obtained between the initial neutral system and the system obtained via the adjoint abstract state operator. The characterization of the duality between controllability and observability is deduced, and then observability conditions are obtained.
For abstract linear systems in Hilbert spaces we revisit the problems of exact controllability and complete stabilizability (stabilizability with an arbitrary decay rate), the latter property is equivalent to exact null controllability. We extend this result to the case when the feedback is not bounded. This enables the characterization of exact null controllability and complete stabilizability for neutral type systems. By duality, we obtain a result about continuous final observability.Illustrative examples are given.
In the present paper we consider controllability and observability of second order linear time invariant systems in matrix form. Without reducing into first order systems we show how the classical conditions for first order linear systems can be generalized to this case. In term of Kalman type criterions these concepts are investigated for second order discrete and continuous time linear systems. It should be pointed out that by repeated differentiation of state and output vector-functions we derive two different systems of linear algebraic equations. Then the initial values x_0, x_1 and input…
The aim of this short paper is to explore a new connection between a conjecture concerning sharp boundary observability estimates for the 1-D heat equation in small time and a conjecture concerning the cost of null-controllability for a 1-D convection-diffusion equation with constant coefficients controlled on the boundary in the vanishing viscosity limit, in the spirit of what is done in [Pierre Lissy, A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, C. R. Math. Acad. Sci. Paris, Volume 352, 2012]. We not…
This paper provides the extension of the observability rank condition and the extension of the controllability rank condition to time-varying nonlinear systems. Previous conditions to check the state observability and controllability, only account for nonlinear systems that do not explicitly depend on time, or, for time-varying systems, they only account for the linear case. In this paper, the general analytic conditions are provided. The paper shows that both these two new conditions (the extended observability rank condition and the extended controllability rank condition) reduce to the well…
We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
We establish the null controllability for linear stochastic fourth order parabolic equations. Utilizing the duality argument, the null controllability is reduced to the observability for backward fourth order stochastic parabolic equations, and the desired observability estimate is obtained by a new global Carleman estimate. Our Carleman estimate is based on a new fundamental identity for a stochastic fourth order parabolic operator.
In this paper, the controllability and observability of linear multi-agent systems over matrix-weighted signed networks are analyzed. Firstly, the definition of equitable partition of matrix-weighted signed multi-agent system is given, and the upper bound of controllable subspace and a necessary condition of controllability are obtained by combining the restriction conditions of the coefficient matrix and matrix weight for the case of fixed and switching topologies, respectively.The influence of different selection methods of coefficient matrices on the results is discussed. Secondly, for the …
We provide algorithmically verifiable necessary and sufficient conditions for fundamental system theoretic properties of discrete time linear systems subject to data losses. More precisely, the systems in our modeling framework are subject to disruptions (data losses) in the feedback loop, where the set of possible data loss sequences is captured by an automaton. As such, the results are applicable in the context of shared (wireless) communication networks and/or embedded architectures where some information on the data loss behaviour is available a priori. We propose an algorithm for deciding…
We develop a linear systems theory that coincides with the existing theories for continuous and discrete dynamical systems, but that also extends to linear systems defined on nonuniform time domains. The approach here is based on generalized Laplace transform methods (e.g. shifts and convolution) from our recent work \cite{DaGrJaMaRa}. We study controllability in terms of the controllability Gramian and various rank conditions (including Kalman's) in both the time invariant and time varying settings and compare the results. We also explore observability in terms of both Gramian and rank condit…
This paper is addressed to studying the exact controllability for stochastic Schrödinger equations by two controls. One is a boundary control in the drift term and the other is an internal control in the diffusion term. By means of the standard duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.
We consider a fully-discrete approximations of 1-D heat equation with dynamic boundary conditions for which we provide a controllability result. The proof of this result is based on a relaxed observability inequality for the corresponding adjoint system. This is done by using a suitable Carleman estimate for such models where the discrete parameters $h$ and $Δt$ are connected to the one of the large Carleman parameters.
We introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions. Approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a Hilbert space is studied. The results are obtained by using the fractional power of operators, multi-valued analysis, and Sadovskii's fixed point theorem. Main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable. As an example, a fractional partial nonlocal control functional …
In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau's paper \cite{DehmanLebeau09} to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time.…
This paper is addressed to a study of the null controllability for the semilinear parabolic equation with a complex principal part. For this purpose, we establish a key weighted identity for partial differential operators $(\a+i\b)\pa_t+\sum\limits_{j,k=1}^n\pa_k(a^{jk}\pa_j)$ (with real functions $\a$ and $\b$), by which we develop a universal approach, based on global Carleman estimate, to deduce not only the desired explicit observability estimate for the linearized complex Ginzburg-Landau equation, but also all the known controllability/observability results for the parabolic, hyperbolic, …
The goal of this article is to present a local exact controllability result for the 2 and 3-dimensional compressible Navier-Stokes equations on a constant target trajectory when the controls act on the whole boundary. Our study is then based on the observability of the adjoint system of some linearized version of the system, which is analyzed thanks to a subsystem for which the coupling terms are somewhat weaker. In this step, we strongly use Carleman estimates in negative Sobolev spaces.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
