# American Institute of Mathematical Sciences

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2163-2480

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## Evolution Equations & Control Theory

September 2018 , Volume 7 , Issue 3

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2018, 7(3): 335-351 doi: 10.3934/eect.2018017 +[Abstract](1773) +[HTML](282) +[PDF](431.8KB)
Abstract:

We study the wave equation on an unbounded network of \begin{document} $N, N∈\mathbb{N}^*$ \end{document}, finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges. The dissipation is given by a damping constant $α>0$ via the condition \begin{document} $\sum_{j = 0}^N\partial_xu_j(0, t) = α \partial_tu_0(0, t)$ \end{document}. We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for \begin{document} $α\not = N+1$ \end{document} we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.

2018, 7(3): 353-371 doi: 10.3934/eect.2018018 +[Abstract](1214) +[HTML](228) +[PDF](478.17KB)
Abstract:

In a Hilbert space \begin{document} $\mathcal H$ \end{document}, we study the convergence properties when \begin{document} $t→+∞$ \end{document} of the trajectories of the second-order differential equation

where \begin{document} $\nablaΦ$ \end{document} is the gradient of a convex continuously differentiable function \begin{document} $Φ: \mathcal H→\mathbb R$ \end{document}, and \begin{document} $γ(t)$ \end{document} is a time-dependent positive viscous damping coefficient. This study aims to offer a unifying vision on the subject, and to complement the article by Attouch and Cabot (J. Diff. Equations, 2017). We obtain convergence rates for the values \begin{document} $Φ(x(t))-{\rm{inf}}_\mathcal{H} Φ$ \end{document} and the velocities under general conditions involving only \begin{document} $γ(·)$ \end{document} and its derivatives. In particular, in the case \begin{document} $γ(t) = \frac{α}{t}$ \end{document}, which is directly connected to the Nesterov accelerated gradient method, our approach allows us to cover all the positive values of \begin{document} $α$ \end{document}, including the subcritical case \begin{document} $α<3$ \end{document}. Our approach is based on the introduction of a new class of Lyapunov functions.

2018, 7(3): 373-401 doi: 10.3934/eect.2018019 +[Abstract](1128) +[HTML](234) +[PDF](574.93KB)
Abstract:

We consider the regularized Tikhonov-like dynamical equilibrium problem: find \begin{document} $u: [0, +∞ [\to\mathcal H$ \end{document} such that for a.e. \begin{document} $t \ge 0$ \end{document} and every \begin{document} $y∈K$ \end{document}, \begin{document} $\langle \dot{u}(t), y-u(t)\rangle +F(u(t), y)+\varepsilon(t) \langle u(t), y-u(t)\rangle \ge 0$ \end{document}, where \begin{document} $F:K×K \to \mathbb{R}$ \end{document} is a monotone bifunction, \begin{document} $K$ \end{document} is a closed convex set in Hilbert space \begin{document} $\mathcal H$ \end{document} and the control function \begin{document} $\varepsilon(t)$ \end{document} is assumed to tend to 0 as \begin{document} $t \to +∞$ \end{document}. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt <∞$ \end{document}, we obtain weak ergodic convergence of \begin{document} $u(t)$ \end{document} to \begin{document} $x∈K$ \end{document} solution of the following equilibrium problem \begin{document} $F(x, y) \ge 0, \;\forall y∈K$ \end{document}. If in addition the bifunction is assumed demipositive, we show weak convergence of \begin{document} $u(t)$ \end{document} to the same solution. By using a slow control \begin{document} $\int_{0}^{+∞} \varepsilon (t) dt = ∞$ \end{document} and assuming that the bifunction \begin{document} $F$ \end{document} is 3-monotone, we show that the term \begin{document} $\varepsilon (t)u(t)$ \end{document} asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of \begin{document} $F$ \end{document}. Also, in the case where \begin{document} $\varepsilon$ \end{document} has a slow control property and \begin{document} $\int_{0}^{+∞}\vert \dot{\varepsilon} (t) \vert dt < +∞$ \end{document}, we show that the strong convergence property of \begin{document} $u(t)$ \end{document} is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm \begin{document} $(ProxPA)$ \end{document} by iteration \begin{document} $x_{n+1} = J^{F_n}_{λ_n}(x_n)$ \end{document} where \begin{document} $F_n(x, y) = F(x, y)+\varepsilon_n \langle x, y-x\rangle$ \end{document}, and \begin{document} $\varepsilon_n$ \end{document} is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm \begin{document} $(DProxA)$ \end{document}: \begin{document} $x_{n+1} = J^F_{λ_n} ((1 - λ_n\varepsilon_n)x_n)$ \end{document}. We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.

2018, 7(3): 403-415 doi: 10.3934/eect.2018020 +[Abstract](1428) +[HTML](213) +[PDF](486.31KB)
Abstract:

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters:

where \begin{document}$l>0$\end{document}, the coefficients \begin{document}$ρ(x)>0, \sigma (x)>0$\end{document}, \begin{document}$q(x)≥0$\end{document} in \begin{document}$\left[ {0,l} \right]$\end{document} and \begin{document}$u$\end{document} is the control acting at the end \begin{document}$x=l$\end{document}. We prove that the linearized problem is exactly controllable in any time \begin{document}$T>0$\end{document}. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument. This problem has been studied by Crépeau [Diff. Integ. Equat., 2002] in the case of constant coefficients \begin{document}$ρ\equiv\sigma \equiv q\equiv1$\end{document}.

2018, 7(3): 417-445 doi: 10.3934/eect.2018021 +[Abstract](986) +[HTML](165) +[PDF](455.62KB)
Abstract:

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. Another interesting problem studied in this work is to establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space, under certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set.

2018, 7(3): 447-463 doi: 10.3934/eect.2018022 +[Abstract](1032) +[HTML](172) +[PDF](408.6KB)
Abstract:

In this paper, we consider the Stokes equations in a two-dimensional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Müntz-Szász Theorem.

2018, 7(3): 465-499 doi: 10.3934/eect.2018023 +[Abstract](1178) +[HTML](175) +[PDF](538.97KB)
Abstract:

In this paper, we establish the Carleman estimates for forward and backward stochastic fourth order Schrödinger equations, on basis of which, we can obtain the observability, unique continuation property and the exact controllability for the forward and backward stochastic fourth order Schrödinger equations.

2018, 7(3): 501-529 doi: 10.3934/eect.2018024 +[Abstract](1129) +[HTML](197) +[PDF](532.41KB)
Abstract:

The present paper studies a new class of problems of optimal control theory with state constraints and second order delay-discrete (DSIs) and delay-differential inclusions (DFIs). The basic approach to solving this problem is based on the discretization method. Thus under the "regularity condition the necessary and sufficient conditions of optimality for problems with second order delay-discrete and delay-approximate DSIs are investigated. Then by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions the sufficient conditions of optimality for delay-DFIs, including the peculiar transversality ones, are proved. Here our main idea is the use of equivalence relations for subdifferentials of Hamiltonian functions and locally adjoint mappings (LAMs), which allow us to make a bridge between the basic optimality conditions of second order delay-DSIs and delay-discrete-approximate problems. In particular, applications of these results to the second order semilinear optimal control problem are illustrated as well as the optimality conditions for non-delayed problems are derived.

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