We give a boundary observability result for a $1$d wave equation with a potential. We then deduce with a Schauder fixed-point argument the existence of a Neumann boundary control for a semi-linear wave equation $\partial_{tt}y - \partial_{xx}y + f(y) = 0$ under an optimal growth assumption at infinity on $f$ of the type $s\ln^2s$. Moreover, assuming additional assumption on $f'$, we construct a minimizing sequence which converges to a control. Numerical experiments illustrate the results. This work extends to the Neumann boundary control case the work of Zuazua in $1993$ and the work of Münch and Trélat in $2022$.
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…
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.
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 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…
Observability inequalities on lattice points are established for non-negative solutions of the heat equation with potentials in the whole space. As applications, some controllability results of heat equations are derived by the above-mentioned observability inequalities.
In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}ρ\left(kw_x\right)_x+γw$, $\left.\left(\sqrt{\frac{k}ρ}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $ρ$, $k$, $γ$ is applied. The results are illustrated by examples.
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…
An application of a conjecture due to Ervedoza and Zuazua concerning the observability of the heat equation in small time to a conjecture due to Coron and Guerrero concerning the uniform controllability of a convection-diffusion equation
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 consider two dimensional Grushin Schrödinger equation posed on a finite cylinder $Ω=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time $T_*>0$ depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at $x=0$, we also show that the observability fails for any $T\leq T_*$.
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.
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.…
We consider the semilinear heat equation posed on a smooth bounded domain $Ω$ of $\mathbb{R}^{N}$ with Dirichlet or Neumann boundary conditions. The control input is a source term localized in some arbitrary nonempty open subset $ω$ of $Ω$. The goal of this paper is to prove the uniform large time global null-controllability for semilinearities $f(s) = \pm |s| \log^α(2+|s|)$ where $α\in [3/2,2)$ which is the case left open by Enrique Fernandez-Cara and Enrique Zuazua in 2000. It is worth mentioning that the free solution (without control) can blow-up. First, we establish the small-time global …
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.
We build up an asymptotic observability identity for the heat equation in the whole space. It says that one can approximately recover a solution, through observing it over some countable lattice points in the space and at one time. This asymptotic identity is a natural extension of the well-known Shannon-Whittaker sampling theorem \cite{Shannon,Whittaker}. According to it, we obtain a kind of feedback null approximate controllability for impulsively controlled heat equations. We also obtain a weak asymptotic observability identity with finitely many observation lattice points. This identity ho…
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.
