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Discrete & Continuous Dynamical Systems - S

2011 , Volume 4 , Issue 3

Issue on new trends in direct, inverse, and control problems for evolution equations

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Preface
Piermarco Cannarsa, Cecilia Cavaterra, Angelo Favini, Alfredo Lorenzi and Elisabetta Rocca
2011, 4(3): i-ii doi: 10.3934/dcdss.2011.4.3i +[Abstract](384) +[PDF](34.0KB)
Abstract:
Science, engineering and economics are full of situations in which one observes the evolution of a given system in time. The systems of interest can differ a lot in nature and their description may require finitely many, as well as infinitely many, variables. Nevertheless, the above models can be formulated in terms of evolution equations, a mathematical structure where the dependence on time plays an essential role. Such equations have long been the object of intensive theoretical study as well as the source of an enormous number of applications.
   A typical class of problems that have been addressed over the years is concerned with the well-posedness of an evolution equation with given initial and boundary conditions (the so-called direct problems). In several applied situations, however, initial conditions are hard to know exactly while measurements of the solution at different stages of its evolution might be available. Different techniques have been developed to recover, from such pieces of information, specific parameters governing the evolution such as forcing terms or diffusion coefficients. The whole body of results in this direction is usually referred to as inverse problems. A third way to approach the subject is to try to influence the evolution of a given system through some kind of external action called control. Control problems may be of very different nature: one may aim at bringing a given system to a desired configuration in finite or infinite time (positional control), or rather try to optimize a performance criterion (optimal control).

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Regularity of boundary traces for a fluid-solid interaction model
Francesca Bucci and Irena Lasiecka
2011, 4(3): 505-521 doi: 10.3934/dcdss.2011.4.505 +[Abstract](437) +[PDF](417.3KB)
Abstract:
We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system-of interest in its own right-establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lamé system.
Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces
Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot and Ahmed Medeghri
2011, 4(3): 523-538 doi: 10.3934/dcdss.2011.4.523 +[Abstract](441) +[PDF](368.5KB)
Abstract:
In this paper we prove some new results concerning a complete abstract second-order differential equation with general Robin boundary conditions. The study is developped in UMD spaces and uses the celebrated Dore-Venni Theorem. We prove existence, uniqueness and maximal regularity of the strict solution. This work completes previous one [3] by authors; see also [11].
Direct and inverse problems in age--structured population diffusion
Gabriella Di Blasio and Alfredo Lorenzi
2011, 4(3): 539-563 doi: 10.3934/dcdss.2011.4.539 +[Abstract](388) +[PDF](421.0KB)
Abstract:
An identification problem for a class of ultraparabolic equations with a non local boundary condition, arising from age-dependent population diffusion, is analized. For such problems existence and uniqueness results as well as continuous dependence upon the data are proved. Regularity results with respect to space variables are also proved, using the theory of parabolic equations in $L^1$-spaces.
Isotropic-nematic phase transitions in liquid crystals
Mauro Fabrizio, Claudio Giorgi and Angelo Morro
2011, 4(3): 565-579 doi: 10.3934/dcdss.2011.4.565 +[Abstract](527) +[PDF](348.0KB)
Abstract:
The paper derives the evolution equations for a nematic liquid crystal, under the action of an electromagnetic field, and characterizes the transition between the isotropic and the nematic state. The non-simple character of the continuum is described by means of the director, of the degree of orientation and their space and time derivatives. Both the degree of orientation and the director are regarded as internal variables and their evolution is established by requiring compatibility with the second law of thermodynamics. As a result, admissible forms of the evolution equations are found in terms of appropriate terms arising from a free-enthalpy potential. For definiteness a free-enthalpy is then considered which provides directly the dielectric and magnetic anisotropies. A characterization is given of thermally-induced transitions with the degree of orientation as a phase parameter.
Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces
Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein and Silvia Romanelli
2011, 4(3): 581-593 doi: 10.3934/dcdss.2011.4.581 +[Abstract](481) +[PDF](127.4KB)
Abstract:
Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$loc$(0,1)\ :\ A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.
Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces
Angelo Favini and Yakov Yakubov
2011, 4(3): 595-614 doi: 10.3934/dcdss.2011.4.595 +[Abstract](385) +[PDF](198.1KB)
Abstract:
We prove an isomorphism of nonlocal boundary value problems for higher order ordinary differential-operator equations generated by one operator in UMD Banach spaces in appropriate Sobolev and interpolation spaces. The main condition is given in terms of $\R$-boundedness of some families of bounded operators generated by the resolvent of the operator of the equation. This implies maximal $L_p$-regularity for the problem. Then we study Fredholmnees of more general problems, namely, with linear abstract perturbation operators both in the equation and boundary conditions. We also present an application of obtained abstract results to boundary value problems for higher order elliptic partial differential equations.
Stability of solutions for nonlinear wave equations with a positive--negative damping
Genni Fragnelli and Dimitri Mugnai
2011, 4(3): 615-622 doi: 10.3934/dcdss.2011.4.615 +[Abstract](362) +[PDF](319.2KB)
Abstract:
We prove a stability result for damped nonlinear wave equations, when the damping changes sign and the nonlinear term satisfies a few natural assumptions.
Kolmogorov equations perturbed by an inverse-square potential
Gisèle Ruiz Goldstein, Jerome A. Goldstein and Abdelaziz Rhandi
2011, 4(3): 623-630 doi: 10.3934/dcdss.2011.4.623 +[Abstract](409) +[PDF](318.4KB)
Abstract:
In this paper we present a nonexistence result of exponentially bounded positive solutions to a parabolic equation of Kolmogorov type with a more general drift term perturbed by an inverse square potential. This result generalizes the one obtained in [8]. Next we introduce some classes of nonlinear operators, related to the filtration operators and the $p$-Laplacian, and involving Kolmogorov operators. We establish the maximal monotonicity of some of these operators. In the third part we discuss the possibility of some nonexistence results in the context of singular potential perturbations of these nonlinear operators.
Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map
Victor Isakov
2011, 4(3): 631-640 doi: 10.3934/dcdss.2011.4.631 +[Abstract](445) +[PDF](341.6KB)
Abstract:
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovery of the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map, when frequency (energy level) is growing. These bounds hold under certain a-priori bounds on the unknown coefficient. Proofs use complex- and real-valued geometrical optics solutions. We outline open problems and possible future developments.
A time reversal based algorithm for solving initial data inverse problems
Kazufumi Ito, Karim Ramdani and Marius Tucsnak
2011, 4(3): 641-652 doi: 10.3934/dcdss.2011.4.641 +[Abstract](506) +[PDF](589.0KB)
Abstract:
We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schrödinger and the Maxwell equations in a bounded domain $\Omega$. We assume that the only available information is a distributed observation (i.e. partial observation of the solution on a sub-domain $\omega$ during a finite time interval $(0,\tau)$). Under some quite natural assumptions (essentially : the exact observability of the system for some time $\tau_{obs}>0$, $\tau\ge \tau_{obs}$ and the existence of a time-reversal operator for the problem), an iterative algorithm based on a Neumann series expansion is proposed. Numerical examples are presented to show the efficiency of the method.
Convergence of solutions of a non-local phase-field system
Stig-Olof Londen and Hana Petzeltová
2011, 4(3): 653-670 doi: 10.3934/dcdss.2011.4.653 +[Abstract](442) +[PDF](423.7KB)
Abstract:
We show that solutions of a two-phase model involving a non-local interactive term separate from the pure phases from a certain time on, even if this is not the case initially. This result allows us to apply a generalized Lojasiewicz-Simon theorem and to establish the convergence of solutions to a single stationary state as time goes to infinity.
An identification problem for a linear evolution equation in a Banach space and applications
Alfredo Lorenzi and Ioan I. Vrabie
2011, 4(3): 671-691 doi: 10.3934/dcdss.2011.4.671 +[Abstract](480) +[PDF](419.8KB)
Abstract:
In this paper we prove both the existence and uniqueness of a solution to an identification problem for a first order linear differential equation in a general Banach space. Namely, we extend the explicit representation for the solution of this problem previously obtained by Anikonov and Lorenzi [1] to the case of an infinitesimal generator of an analytic $C_0$-semigroup of contractions to the general nonanalytic case and also to the case of a restriction expressed in terms of an operator-valued measure. So, our abstract result handles both parabolic and hyperbolic equations and systems.
Exponential stability of the wave equation with boundary time-varying delay
Serge Nicaise, Cristina Pignotti and Julie Valein
2011, 4(3): 693-722 doi: 10.3934/dcdss.2011.4.693 +[Abstract](535) +[PDF](505.7KB)
Abstract:
We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain $\Omega\subset\RR^n.$ Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.
Linear evolution equations with strongly measurable families and application to the Dirac equation
Noboru Okazawa and Kentarou Yoshii
2011, 4(3): 723-744 doi: 10.3934/dcdss.2011.4.723 +[Abstract](400) +[PDF](296.7KB)
Abstract:
A new existence and uniqueness theorem is established for linear evolution equations of hyperbolic type with strongly measurable coefficients in a separable Hilbert space. The result is applied to the Dirac equation with time-dependent potential.
Riesz systems, spectral controllability and a source identification problem for heat equations with memory
Luciano Pandolfi
2011, 4(3): 745-759 doi: 10.3934/dcdss.2011.4.745 +[Abstract](446) +[PDF](340.0KB)
Abstract:
In this paper we show that recent results on a Riesz basis associated to a heat equation with memory can be used in order to solve a source identification problem.
Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems
Judith Vancostenoble
2011, 4(3): 761-790 doi: 10.3934/dcdss.2011.4.761 +[Abstract](424) +[PDF](514.6KB)
Abstract:
We consider the following class of degenerate/singular parabolic operators:

$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,

associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters $\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities

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