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Evolution Equations & Control Theory

2014 , Volume 3 , Issue 1

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Existence and asymptotic behaviour for solutions of dynamical equilibrium systems
Zaki Chbani and  Hassan Riahi
2014, 3(1): 1-14 doi: 10.3934/eect.2014.3.1 +[Abstract](45) +[PDF](394.9KB)
In this paper, we give an existence result for the following dynamical equilibrium problem: $\langle \frac{du}{dt},v-u(t)\rangle+F(u(t),v)\geq 0 \;\; \forall v\in K $ and for $a.e. \;t \geq 0$, where $K$ is a closed convex set in a Hilbert space and $ F:K \times K \rightarrow \mathbb{R}$ is a monotone bifunction. We introduce a class of demipositive bifunctions and use it to study the asymptotic behaviour of the solution $ u(t) $ when $ t\rightarrow\infty $. We obtain weak convergence of $ u(t) $ to some solution $x\in K$ of the equilibrium problem $F(x,y)\geq 0 $ for every $y\in K$. Our applications deal with the asymptotic behaviour of the dynamical convex minimization and dynamical system associated to saddle convex-concave bifunctions. We then present a new neural model for solving a convex programming problem.
Well-posedness for degenerate Schrödinger equations
Massimo Cicognani and  Michael Reissig
2014, 3(1): 15-33 doi: 10.3934/eect.2014.3.15 +[Abstract](40) +[PDF](461.6KB)
We consider the initial value problem for Schrödinger type equations $$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.
Optimal control for stochastic heat equation with memory
Fulvia Confortola and  Elisa Mastrogiacomo
2014, 3(1): 35-58 doi: 10.3934/eect.2014.3.35 +[Abstract](41) +[PDF](455.5KB)
In this paper, we investigate the existence and uniqueness of solutions for a class of evolutionary integral equations perturbed by a noise arising in the theory of heat conduction. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.
Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.
Sébastien Court
2014, 3(1): 59-82 doi: 10.3934/eect.2014.3.59 +[Abstract](29) +[PDF](527.9KB)
This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential decay rate, by a boundary deformation velocity which can be chosen in the form of a feedback operator. We then show that this boundary feedback operator can be obtained from an internal deformation of the solid which satisfies the linearized physical constraints that a self-propelled solid has to satisfy.
Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.
Sébastien Court
2014, 3(1): 83-118 doi: 10.3934/eect.2014.3.83 +[Abstract](36) +[PDF](584.2KB)
In this second part we prove that the full nonlinear fluid-solid system introduced in Part I is stabilizable by deformations of the solid that have to satisfy nonlinear constraints. Some of these constraints are physical and guarantee the self-propelled nature of the solid. The proof is based on the boundary feedback stabilization of the linearized system. From this boundary feedback operator we construct a deformation of the solid which satisfies the aforementioned constraints and which stabilizes the nonlinear system. The proof is made by a fixed point method.
Recovery of time dependent volatility coefficient by linearization
Victor Isakov
2014, 3(1): 119-134 doi: 10.3934/eect.2014.3.119 +[Abstract](32) +[PDF](380.3KB)
We study the problem of reconstruction of special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator $W$ which is linear in perturbation of volatility. We further simplify the linearized inverse problem and obtain unique solvability result in basic functional spaces. By using the Laplace transform in time we simplify the kernels of integral operators for $W$ and we obtain uniqueness and stability results for volatility under natural condition of smallness of the spacial interval where one prescribes the (market) data. We propose a numerical algorithm based on our analysis of the linearized problem.
Cross-like internal observability of rectangular membranes
Vilmos Komornik and  Bernadette Miara
2014, 3(1): 135-146 doi: 10.3934/eect.2014.3.135 +[Abstract](33) +[PDF](334.4KB)
We present a new way to establish internal observability results for the wave equation. Our method is based on some variants of Ingham's theorem on nonharmonic Fourier series, due to Loreti, Valente and Mehrenberger.
Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method
Evrad M. D. Ngom , Abdou Sène and  Daniel Y. Le Roux
2014, 3(1): 147-166 doi: 10.3934/eect.2014.3.147 +[Abstract](43) +[PDF](492.2KB)
In this work we study the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$, around a given steady-state flow, by means of a boundary control. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions. The resulting feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.
Boundary approximate controllability of some linear parabolic systems
Guillaume Olive
2014, 3(1): 167-189 doi: 10.3934/eect.2014.3.167 +[Abstract](49) +[PDF](502.2KB)
This paper focuses on the boundary approximate controllability of two classes of linear parabolic systems, namely a system of $n$ heat equations coupled through constant terms and a $2 \times 2$ cascade system coupled by means of a first order partial differential operator with space-dependent coefficients.
    For each system we prove a sufficient condition in any space dimension and we show that this condition turns out to be also necessary in one dimension with only one control. For the system of coupled heat equations we also study the problem on rectangle, and we give characterizations depending on the position of the control domain. Finally, we prove the distributed approximate controllability in any space dimension of a cascade system coupled by a constant first order term.
    The method relies on a general characterization due to H.O. Fattorini.
Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains
Yutaka Tsuzuki
2014, 3(1): 191-206 doi: 10.3934/eect.2014.3.191 +[Abstract](33) +[PDF](229.5KB)
This paper is concerned with the system of nonlinear heat equations with constraints coupled with Navier-Stokes equations in two-dimensional domains. In 2012, Sobajima, Tsuzuki and Yokota proved the existence and uniqueness of solutions to the system with heat equations including the diffusion term $\Delta\theta$, where $\theta$ represents the temperature. This paper gives the existence result in which the Laplace operator $\Delta$ is replaced with the $p$-Laplace operator $\Delta\rho$, where $p>2$.

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