This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4, 5, 6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems....

This paper deals with the observability analysis and the observer synthesis of a class of nonlinear systems. In the single output case, it is known [4-6] that systems which are observable independently of the inputs, admit an observable canonical form. These systems are called uniformly observable systems. Moreover, a high gain observer for these systems can be designed on the basis of this canonical form. In this paper, we extend the above results to multi-output uniformly observable systems....

On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11, 18, 22, 26, 27, 38, 40] and other references therein).

On an arbitrary reflexive Banach space, we build asymptotic observers for an abstract class of nonlinear control systems with possible compact outputs. An important part of this paper is devoted to various examples, where we discuss the existence of persistent inputs which make the system observable. These results make a wide generalization to a nonlinear framework of previous works on the observation problem in infinite dimension (see [11,18,22,26,27,38,40] and other references therein).

The focus of this paper is to develop reliable observer and filtering techniques for finite-dimensional battery models that adequately describe the charging and discharging behaviors. For this purpose, an experimentally validated battery model taken from the literature is extended by a mathematical description that represents parameter variations caused by aging. The corresponding disturbance models account for the fact that neither the state of charge, nor the above-mentioned parameter variations...

We are concerned with the null controllability of a linear coupled population dynamics system or the so-called prey-predator model with Holling type I functional response of predator wherein both equations are structured in age and space. It is worth mentioning that in our case, the space variable is viewed as the “gene type” of population. The studied system is with two different dispersion coefficients which depend on the gene type variable and degenerate in the boundary. This system will be governed...

In this paper, we prove the exact null controllability of certain diffusion system by rewriting it as an equivalent nonlinear parabolic integrodifferential equation with variable coefficients in a bounded interval of $ℝ$ with a distributed control acting on a subinterval. We first prove a global null controllability result of an associated linearized integrodifferential equation by establishing a suitable observability estimate for adjoint system with appropriate assumptions on the coefficients. Then...

The goal of this note is to present the results of the references [5] and [4]. We study the null controllability of the parabolic equations associated with the Grushin-type operator ${\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_y^2$ ($\gamma \>0$) in the rectangle $(x,y)\in (-1,1)\times (0,1)$ or with the Kolmogorov-type operator $v^{\gamma }{\partial }_{x}f+{\partial }_v^{2}f$ ($\gamma \in \{1,2\}$) in the rectangle $(x,v)\in 𝕋\times (-1,1)$, under an additive control supported in an open subset $\omega$ of the space domain.We prove that the Grushin-type equation is null controllable in any positive time for $\gamma \<1$ and that there is no time for which it is null controllable for $\gamma \>1$....

We study the null controllability of the parabolic equation associated with the Grushin-type operator $A={\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_{\gamma }^2,(\gamma >0)$, in the rectangle $\Omega =(-1,1)\times (0,1)$, under an additive control supported in an open subset $\omega$ of $\Omega$. We prove that the equation is null controllable in any positive time for $\gamma <1$ and that there is no time for which it is null controllable for $\gamma >1$. In the transition regime $\gamma =1$ and when $\omega$ is a strip $\omega =(a,b)\times (0,1)(0<a,b\le 1)$), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular...

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^N$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_n{\omega }_n$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_n$, we prove that every initial datum in some weighted $L^2$ space can be controlled to zero by usual control functions, and every initial datum in $L^2\left(\Omega \right)$ can...

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^N$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_n{\omega }_n$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_n$, we prove that every initial datum in some weighted L2 space can be controlled to zero by usual control functions, and every initial datum in L2(Ω)...

Basic properties on linearization by output injection are investigated in this paper. A special structure is sought which is linear up to a suitable output injection and under a suitable change of coordinates. It is shown how an observer may be designed using theory available for linear time delay systems.

This paper presents two observability inequalities for the heat equation over $\Omega \times (0,T)$. In the first one, the observation is from a subset of positive measure in $\Omega \times (0,T)$, while in the second, the observation is from a subset of positive surface measure on $\partial \Omega \times (0,T)$. It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

We consider smooth single-input, two-output systems on a compact manifold X. We show that the set of systems that are observable for any polynomial input whose degree is less than or equal to a given bound contains an open and dense subset of the set of smooth systems.

Observability of a general nonlinear system—given in terms of an ODE $\dot{x}=f\left(x\right)$ and an output map $y=c\left(x\right)$—is defined as in linear system theory (i.e. if $f\left(x\right)=Ax$ and $c\left(x\right)=Cx$). In contrast to standard treatment of the subject we present a criterion for observability which is not a generalization of a known linear test. It is obtained by evaluation of “approximate first integrals”. This concept is borrowed from nonlinear control theory where it appears under the label “Dissipation Inequality” and serves as a link with Hamilton-Jacobi...