Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.     

Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

q-Difference linear control systems

We propose q-difference versions of some basic results of complete controllability and observability of linear systems such as criterions of controllability and observability, controllability and observability canonical forms, the q-duality theorem, the interconnection between these concepts and that of polynomials without common zeros. As an appendix, is given a simple but computationally meaningful Maple Procedure for the controllability criterion for time constant systems.     

Simultaneous controllability of damped wave equations

We study the exact controllability of q uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Upper bounds of a stabilizer order for linear finite-dimensional systems

The problem of stabilizing linear dynamic systems by a stabilizer (a dynamic system) is considered. The upper bounds of a stabilizer order obtained using two Hidenori Kimura results are studied. The bound k ₀ is shown to be better than the bounds k ₁ and k ₂ only in one case. In addition, all possible relations between three bounds k ₀, k ₁, and k ₂ are proven to be realized in the space of parameters of observability and controllability indices, i.e., there is a dynamic system with the respective observability and controllability indices.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of R^n, with a controller w to be any given nonempty open subset of Ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T 〉 0 when the control is restricted to be active in ω× E, where E is any given subset of [0, T] with positive （Legesgue） measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.     

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations t_∂±x_∂(cx_∂) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.     

QUINTESSENTIAL PRIMES AND IDEAL TOPOLOGIES OVER A MODULE

Let I be an ideal of a Noetherian ring R, N a finitely generated R-module and let S be a multiplicatively closed subset of R. We define the Nth (S)-symbolic power of I w.r.t. N as S(I ⁿ N) = ∪ _s∈S (I ⁿ N: _N s). The purpose of this paper is to show that the topologies defined by {Iⁿ N} _n≥0 and {S(Iⁿ N)} _n≥0 are equivalent (resp. linearly equivalent) if and only if S is disjoint from the quintessential (resp. essential) primes of I w.r.t. N.     

Controllability of systems of Stokes equations with one control force: existence of insensitizing controls

In this paper we establish some exact controllability results for systems of two parabolic equations of the Stokes kind. In a first part, we prove the existence of insensitizing controls for the L² norm of the solutions and the curl of solutions of linear Stokes equations. Then, in the limit case where one can expect null controllability to hold for a system of two Stokes equations (namely, when the coupling terms concern first and second order derivatives, respectively), we prove this result for some general couplings.     

A capacitance matrix method for Dirichlet problem on polygon region

Summary An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN ²log₂ N (N=1/h) where the cost of solving the capacitance matrix equations isNlog₂ N on regular grids andN ^3/2log₂ N on irregular ones.     

Expanding the additive reduct of a model of Peano arithmetic

Let M be a model of first order Peano arithmetic ( PA ) and I an initial segment of M that is closed under multiplication. LetM₀ be the {0, 1,+}‐reduct ofM. We show that there is another model N of PA that is also an expansion of M₀ such that a · ^M a = a · ^N a if and only if a ∈ I for all a ∈ M.     

A characterization of Cohen-Macaulay modules and local cohomology

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by height_N I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included. Received: 3 July 2005     

Global continuation for first order systems over the half‐line involving parameters

Let X be one of the functional spaces W^1,p ((0, ∞), ?^N ) or C₀¹ ([0, ∞), ?^N ), we study the global continuation in λ for solutions (λ, u, ξ) ∈ ? × X × ?^k of the following system of ordinary differential equations: where ?^N = X₁ ⊕ X₂ is a given decomposition, with associated projection P: ?^N → X₁. Under appropriate conditions upon the given functions F and φ, this problem gives rise to a nonlinear Fredholm operator which is proper on the closed bounded subsets of ? × X × ?^k and whose zeros correspond to the solutions of the original problem. Using a new abstract continuation result, based on a recent degree theory for proper Fredholm mappings of index zero, we reduce the continuation problem to that of finding a priori estimates for the possible solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

含有p拉普拉斯算子方程的解的存在性研究

By using the perturbation theories on sums of ranges for nonlinear accretive mappings of Calvert and Gupta (1978), the abstract result on the existence of a solution u ∈ Lp (Ω) to nonlinear equations involving p-Laplacian operator △p, where 2N/N 1 ＜ p ＜ ∞ and N (≥ 1 )denotes the dimension of RN,is studied. The equation discussed and the methods shown in the paper are continuation and complement to the corresponding results of Li and Zhen's previous papers. To obtain the result ,some new techniques are used.     

Controllability of nonlinear systems

Some sufficient conditions are presented for the controllability of general nonlinear systems. First, the controllability problem is transformed into the problem of existence of fixed points for some operator; using Schauder's theorem, it is derived that a sufficient condition for controllability is the existence of a subsetS inC _n+m (T) which is invariant for a derived operator. Secondly, with the aid of the notion of comparison principle, the existence of the subsetS is guaranteed by the existence of solutions for some nonlinear integral inequality or equality equations. For example, one solution for such nonlinear integral equations is obtained under the assumption of the uniform boundedness for a nonlinear term of the differential equation.     

A weighted identity for partial differential operators of second order and its applications

In this Note, a weighted identity for partial differential operators of second order is established. As its applications, one may deduce all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimate. Meanwhile, a new controllability/observability result is presented for the parabolic equations with a complex principal part. To cite this article: X. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Admissibility and Observability of Observation Operators for Semilinear Problems

This paper deals with semilinear evolution equations with unbounded observation operators. Sufficient conditions are given guaranteeing that the output function of a semilinear system is in L²_loc([0, ∞); Y). We prove that the Lebesgue extension of the observation operators are invariant under nonlinear globally Lipschitz continuous perturbations. Further, relations between the corresponding -extensions are studied. We show that exact observability of linear autonomous system is conserved under small Lipschitz perturbations. The obtained results are illustrated by several examples.      

Null controllability in function space of nonlinear retarded systems with limited control

The question of controlling both linear and nonlinear retarded functional differential equations from an initial function in a Sobolev space W₂⁽¹⁾ to zero in W₂⁽¹⁾ is considered when the controls are square integrable with values in a compact subset U of m-dimensional Euclidean space with zero in its interior. Sufficient computable criteria for null controllability are given.