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Mathematical Control & Related Fields

December 2021 , Volume 11 , Issue 4

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Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains
Ji Shu, Linyan Li, Xin Huang and Jian Zhang
2021, 11(4): 715-737 doi: 10.3934/mcrf.2020044 +[Abstract](1015) +[HTML](408) +[PDF](429.96KB)
Abstract:

We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in \begin{document}$ \mathbb{R}^{3} $\end{document} are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.

First order necessary conditions of optimality for the two dimensional tidal dynamics system
Manil T. Mohan
2021, 11(4): 739-769 doi: 10.3934/mcrf.2020045 +[Abstract](955) +[HTML](412) +[PDF](437.59KB)
Abstract:

In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.

Strict dissipativity for discrete time discounted optimal control problems
Lars Grüne, Matthias A. Müller, Christopher M. Kellett and Steven R. Weller
2021, 11(4): 771-796 doi: 10.3934/mcrf.2020046 +[Abstract](953) +[HTML](387) +[PDF](447.68KB)
Abstract:

The paradigm of discounting future costs is a common feature of economic applications of optimal control. In this paper, we provide several results for such discounted optimal control aimed at replicating the now well-known results in the standard, undiscounted, setting whereby (strict) dissipativity, turnpike properties, and near-optimality of closed-loop systems using model predictive control are essentially equivalent. To that end, we introduce a notion of discounted strict dissipativity and show that this implies various properties including the existence of available storage functions, required supply functions, and robustness of optimal equilibria. Additionally, for discount factors sufficiently close to one we demonstrate that strict dissipativity implies discounted strict dissipativity and that optimally controlled systems, derived from a discounted cost function, yield practically asymptotically stable equilibria. Several examples are provided throughout.

A stackelberg game of backward stochastic differential equations with partial information
Yueyang Zheng and Jingtao Shi
2021, 11(4): 797-828 doi: 10.3934/mcrf.2020047 +[Abstract](1144) +[HTML](395) +[PDF](426.58KB)
Abstract:

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-\begin{document}$ \sigma $\end{document}-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE. Four high-dimensional Riccati equations are introduced to represent the state estimate feedback for the optimal control of the leader. Theoretic results are applied to a pension fund management problem of two players in the financial market.

Stochastic maximum principle for problems with delay with dependence on the past through general measures
Giuseppina Guatteri and Federica Masiero
2021, 11(4): 829-855 doi: 10.3934/mcrf.2020048 +[Abstract](891) +[HTML](382) +[PDF](382.44KB)
Abstract:

We prove a stochastic maximum principle for a control problem where the state equation is delayed both in the state and in the control, and both the running and the final cost functionals may depend on the past trajectories. The adjoint equation turns out to be a new form of linear anticipated backward stochastic differential equations (ABSDEs in the following), and we prove a direct formula to solve these equations.

Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces
Sumit Arora, Manil T. Mohan and Jaydev Dabas
2021, 11(4): 857-883 doi: 10.3934/mcrf.2020049 +[Abstract](997) +[HTML](396) +[PDF](402.85KB)
Abstract:

In this paper, we investigate the approximate controllability problems of certain Sobolev type differential equations. Here, we obtain sufficient conditions for the approximate controllability of a semilinear Sobolev type evolution system in Banach spaces. In order to establish the approximate controllability results of such a system, we have employed the resolvent operator condition and Schauder's fixed point theorem. Finally, we discuss a concrete example to illustrate the efficiency of the results obtained.

Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions
Ali Wehbe, Rayan Nasser and Nahla Noun
2021, 11(4): 885-904 doi: 10.3934/mcrf.2020050 +[Abstract](993) +[HTML](361) +[PDF](540.81KB)
Abstract:

We investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type \begin{document}$ t^{-1} $\end{document} for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type \begin{document}$ t^{-2/5} $\end{document} if the strip is localized near the boundary. Else, it's of type \begin{document}$ t^{-1/3} $\end{document}. The main novelty in this paper is that the geometric situations covered here are richer and less restrictive than those considered in [31], [28], [19] and include in particular an example where the damping region is localized faraway from the boundary. Note that part of the results of this paper was announced in [22].

Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential
Pierluigi Colli, Gianni Gilardi and Gabriela Marinoschi
2021, 11(4): 905-934 doi: 10.3934/mcrf.2020051 +[Abstract](841) +[HTML](344) +[PDF](430.22KB)
Abstract:

In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential \begin{document}$ \mu $\end{document} is present. This term consists of a subdifferential operator \begin{document}$ S $\end{document} in \begin{document}$ L^2(\Omega) $\end{document} (where \begin{document}$ \Omega $\end{document} is the domain where the evolution takes place) acting on the difference of the phase variable \begin{document}$ \varphi $\end{document} and a given state \begin{document}$ {\varphi^*} $\end{document}, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential \begin{document}$ \mu $\end{document}. Next, by assuming that \begin{document}$ S = \rho\;{\rm{sign}} $\end{document}, a multiple of the \begin{document}$ \;{\rm{sign}} $\end{document} operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for \begin{document}$ \mu $\end{document} and under suitable conditions on \begin{document}$ \rho $\end{document} and \begin{document}$ \Omega $\end{document}, we also prove the sliding mode property, that is, that \begin{document}$ \varphi $\end{document} is forced to join the evolution of \begin{document}$ {\varphi^*} $\end{document} in some time \begin{document}$ T^* $\end{document} lower than the given final time \begin{document}$ T $\end{document}. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on \begin{document}$ \varphi $\end{document}.

Strict dissipativity analysis for classes of optimal control problems involving probability density functions
Arthur Fleig and Lars Grüne
2021, 11(4): 935-964 doi: 10.3934/mcrf.2020053 +[Abstract](807) +[HTML](355) +[PDF](1347.36KB)
Abstract:

Motivated by the stability and performance analysis of model predictive control schemes, we investigate strict dissipativity for a class of optimal control problems involving probability density functions. The dynamics are governed by a Fokker-Planck partial differential equation. However, for the particular classes under investigation involving linear dynamics, linear feedback laws, and Gaussian probability density functions, we are able to significantly simplify these dynamics. This enables us to perform an in-depth analysis of strict dissipativity for different cost functions.

A stability result for the diffusion coefficient of the heat operator defined on an unbounded guide
Laure Cardoulis, Michel Cristofol and Morgan Morancey
2021, 11(4): 965-985 doi: 10.3934/mcrf.2020054 +[Abstract](818) +[HTML](333) +[PDF](475.51KB)
Abstract:

In this article we consider the inverse problem of determining the diffusion coefficient of the heat operator in an unbounded guide using a finite number of localized observations. For this problem, we prove a stability estimate in any finite portion of the guide using an adapted Carleman inequality. The measurements are located on the boundary of a larger finite portion of the guide. A special care is required to avoid measurements on the cross-section boundaries which are inside the actual guide. This stability estimate uses a technical positivity assumption. Using arguments from control theory, we manage to remove this assumption for the inverse problem with a given non homogeneous boundary condition.

2020 Impact Factor: 1.284
5 Year Impact Factor: 1.345
2020 CiteScore: 1.9

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