# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations & Control Theory

September 2020 , Volume 9 , Issue 3

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2020, 9(3): 581-604 doi: 10.3934/eect.2020025 +[Abstract](1277) +[HTML](422) +[PDF](396.33KB)
Abstract:

A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

2020, 9(3): 605-635 doi: 10.3934/eect.2020026 +[Abstract](1221) +[HTML](398) +[PDF](465.07KB)
Abstract:

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanuvilov, is obtained from the global null controllability of the linearized system provided by Lyusternik's Inverse Mapping Theorem. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.

2020, 9(3): 635-672 doi: 10.3934/eect.2020027 +[Abstract](1160) +[HTML](436) +[PDF](1404.68KB)
Abstract:

It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depends on the operator norm of the oblique projection, on the polynomial degree of the nonlinearity, on the norm of the initial condition, and on the total volume covered by the actuators. The range of the feedback controller coincides with the range of the oblique projection, which is the linear span of the actuators. The oblique projection is performed along the orthogonal complement of a subspace spanned by a suitable finite number of eigenfunctions of the diffusion operator. For rectangular domains, it is possible to explicitly construct/place the actuators so that the stability of the closed-loop system is guaranteed. Simulations are presented, which show the semiglobal stabilizing performance of the nonlinear feedback.

2020, 9(3): 673-692 doi: 10.3934/eect.2020028 +[Abstract](1902) +[HTML](391) +[PDF](355.85KB)
Abstract:

Controllability of coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Roughly speaking, the main challenge is controlling a system with less inputs than equations. In this paper this is successfully done for a system of Korteweg-de Vries equations posed on an oriented tree shaped network. The couplings and the controls appear only on boundary conditions.

2020, 9(3): 693-719 doi: 10.3934/eect.2020029 +[Abstract](1108) +[HTML](409) +[PDF](401.73KB)
Abstract:

In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability by using two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.

2020, 9(3): 721-732 doi: 10.3934/eect.2020030 +[Abstract](1133) +[HTML](386) +[PDF](344.54KB)
Abstract:

It is the purpose of this note to investigate the initial value problem for a focusing semi-linear damped Schrödinger equation. Indeed, in the energy sub-critical regime, one obtains global well-posedness and scattering in the energy space, depending on the order of the fractional dissipation.

2020, 9(3): 733-751 doi: 10.3934/eect.2020031 +[Abstract](1337) +[HTML](437) +[PDF](364.35KB)
Abstract:

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on the Azouani, Olson, and Titi (AOT) algorithm, but applied to the 2D Navier-Stokes-Voigt equations. Adapting the AOT algorithm to regularized versions of Navier-Stokes has been done before, but the innovation of this work is to drive the assimilation equation with observational data, rather than data from a regularized system. We first prove that this new system is globally well-posed. Moreover, we prove that for any admissible initial data, the \begin{document}$L^2$\end{document} and \begin{document}$H^1$\end{document} norms of error are bounded by a constant times a power of the Voigt-regularization parameter \begin{document}$\alpha>0$\end{document}, plus a term which decays exponentially fast in time. In particular, the large-time error goes to zero algebraically as \begin{document}$\alpha$\end{document} goes to zero. Assuming more smoothness on the initial data and forcing, we also prove similar results for the \begin{document}$H^2$\end{document} norm.

Qiang Li and
2020, 9(3): 753-772 doi: 10.3934/eect.2020032 +[Abstract](1354) +[HTML](474) +[PDF](343.87KB)
Abstract:

In this paper, we are devoted to consider the periodic problem for the neutral evolution equation with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equation. In addition, with the aid of a new integral inequality with delay, we present essential conditions on the nonlinear function to guarantee that the equation has an asymptotically stable periodic mild solution.

2020, 9(3): 773-793 doi: 10.3934/eect.2020033 +[Abstract](2021) +[HTML](498) +[PDF](559.21KB)
Abstract:

This paper studies a time-fractional diffusion-wave equation with a linear source function. First, some stability results on parameters of the Mittag-Leffler functions are established. Then, we focus on studying the continuity of the solution of both the initial problem and the inverse initial value problems corresponding to the fractional-order in our main results. One of the difficulties encounteblack comes from estimating all constants independently of the fractional orders. Finally, we present some numerical results to confirm the effectiveness of our methods.

2020, 9(3): 795-816 doi: 10.3934/eect.2020034 +[Abstract](1042) +[HTML](362) +[PDF](351.04KB)
Abstract:

Here we study the problem of boundary feedback stabilization to unbounded trajectories for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. The simple-form feedback allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument its stability is proved. The approach requires the initial data to be a random variable implying the fact that the solution of the random equation is not adapted. Thus, one cannot recover the solution of the initial stochastic equation from the random one. Hence, the designed feedback controller stabilizes the associated random equation and not the original stochastic equation. Anyway, it stabilizes its random version.

2020, 9(3): 817-843 doi: 10.3934/eect.2020035 +[Abstract](1039) +[HTML](364) +[PDF](392.23KB)
Abstract:

In this paper, we are concerned with an optimal control problem where the system is driven by fully coupled forward-backward stochastic differential equation of mean-field type with risk-sensitive performance functional. We study the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of the initial stochastic control problem in this type of risk-sensitive performance problem, where an admissible set of controls are convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. Finally, we illustrate our main result of this paper by giving two examples of risk-sensitive control problem under linear stochastic dynamics with exponential quadratic cost function, the second example will be a mean-variance portfolio with a recursive utility functional optimization problem involving optimal control. The explicit expression of the optimal portfolio selection strategy is obtained in the state feedback.

2020, 9(3): 845-863 doi: 10.3934/eect.2020036 +[Abstract](1007) +[HTML](404) +[PDF](381.14KB)
Abstract:

In this paper, we mainly consider a time-varying semi-linear integro-differential inclusion with Clarke sub-differential and a non-local initial condition. By a suitable Green function combined with a resolvent operator, we firstly formulate its mild solutions and show that it admits at least one mild solution which can exist in a well-defined ball with a radius big enough. Through constructing a proper functional, we then derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a sufficient condition for the approximate controllability of the time-varying semi-linear integro-differential inclusion. Lastly, we also consider the finite approximate controllability of the time-varying semi-linear integro-differential inclusion via variational method.

2020, 9(3): 865-889 doi: 10.3934/eect.2020037 +[Abstract](966) +[HTML](319) +[PDF](462.52KB)
Abstract:

We consider the Cauchy problem associated to the fourth-order nonlinear Schrödinger-Hartree equation with variable dispersion coefficients. The variable dispersion coefficients are assumed to be continuous or periodic and piecewise constant in time functions. We prove local and global well-posedness results for initial data in \begin{document}$H^s$\end{document}-spaces. We also analyze the scaling limit of the fast dispersion management and the convergence to a model with averaged dispersions.

2020, 9(3): 891-914 doi: 10.3934/eect.2020038 +[Abstract](1099) +[HTML](360) +[PDF](374.58KB)
Abstract:

Our goal is to establish some sufficient conditions for the solvability of the nonlocal final value problem involving a class of partial differential equations, which describes the anomalous diffusion phenomenon. Our analysis is based on the theory of completely positive functions, resolvent operators and fixed point arguments in suitable function spaces. Especially, utilizing the regularity of resolvent operators, we are able to deal with non-Lipschitz cases. The obtained results, in particular, extend recent ones proved for fractional diffusion equations.

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