# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

June 2021 , Volume 11 , Issue 2

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2021, 11(2): 237-251 doi: 10.3934/mcrf.2020035 +[Abstract](913) +[HTML](267) +[PDF](335.5KB)
Abstract:

In this paper, we consider a parabolic PDE on a torus of arbitrary dimension. The nonlinear term is a smooth function of polynomial growth of any degree. In this general setting, the Cauchy problem is not necessarily well posed. We show that the equation in question is approximately controllable by only a finite number of Fourier modes. This result is proved by using some ideas from the geometric control theory introduced by Agrachev and Sarychev.

2021, 11(2): 253-289 doi: 10.3934/mcrf.2020036 +[Abstract](848) +[HTML](283) +[PDF](528.96KB)
Abstract:

This paper is devoted to the study of a class of optimal control problems governed by 1–D Kobayashi–Warren–Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141–150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi–Warren–Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1–3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.

2021, 11(2): 291-312 doi: 10.3934/mcrf.2020037 +[Abstract](906) +[HTML](266) +[PDF](431.18KB)
Abstract:

In this paper, we first introduce a new spatial-temporal interaction operator to describe the space-time dependent phenomena. Then we consider the stochastic optimal control of a new system governed by a stochastic partial differential equation with the spatial-temporal interaction operator. To solve such a stochastic optimal control problem, we derive an adjoint backward stochastic partial differential equation with spatial-temporal dependence by defining a Hamiltonian functional, and give both the sufficient and necessary (Pontryagin-Bismut-Bensoussan type) maximum principles. Moreover, the existence and uniqueness of solutions are proved for the corresponding adjoint backward stochastic partial differential equations. Finally, our results are applied to study the population growth problems with the space-time dependent phenomena.

2021, 11(2): 313-328 doi: 10.3934/mcrf.2020038 +[Abstract](1008) +[HTML](264) +[PDF](413.4KB)
Abstract:

This work is motivated by recent interest in the topic of pointwise tracking type optimal control problems for the Stokes problem. Pointwise tracking consists of point evaluations in the objective functional which lead to Dirac measures appearing as source terms of the adjoint problem. Considering bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical counterpart. We show a sub-optimal convergence result in three dimensions that nonetheless improves the results known from the literature. Finally, we offer supporting numerical experiments and insights towards optimal approximation error estimates.

2021, 11(2): 329-351 doi: 10.3934/mcrf.2020039 +[Abstract](695) +[HTML](182) +[PDF](452.81KB)
Abstract:

In this paper, we present some properties of time optimal controls for linear ODEs with the ball-type control constraint. More precisely, given an optimal control, we build up an upper bound for the number of its switching points; show that it jumps from one direction to the reverse direction at each switching point; give its dynamic behaviour between two consecutive switching points; and study its switching directions.

2021, 11(2): 353-371 doi: 10.3934/mcrf.2020040 +[Abstract](637) +[HTML](178) +[PDF](397.82KB)
Abstract:

In this paper we study an abstract thermoelastic system in Hilbert space with infinite memory and time delay. Under some suitable conditions, we prove the well-posedness by invoking semigroup theory. Since the damping may stabilize the system while the delay may destabilize it, we discuss the interaction between the damping and the delay term, and obtain that the system is uniformly stable when the effect of damping is greater than that of time delay. By establishing suitable Lyapunov functionals which are equivalent to the energy of system we also establish the general energy decay results for abstract thermoelastic system.

2021, 11(2): 373-401 doi: 10.3934/mcrf.2020041 +[Abstract](734) +[HTML](189) +[PDF](711.91KB)
Abstract:

The local geometry of sub-Finslerian structures in dimension 3 associated with a maximum norm is studied in the contact case. A normal form is given. The short extremals, the local switching, conjugate and cut loci, and the small spheres are described in the generic case.

2021, 11(2): 403-431 doi: 10.3934/mcrf.2020042 +[Abstract](700) +[HTML](192) +[PDF](571.32KB)
Abstract:

This paper deals with an inverse problem for a non-self-adjoint Schrödinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to-Neumann map. We establish in dimension \begin{document}$n\geq2$\end{document}, an Hölder type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schrödinger equation and the use of a Carleman estimate designed for elliptic operators.

2021, 11(2): 433-478 doi: 10.3934/mcrf.2020043 +[Abstract](968) +[HTML](193) +[PDF](519.98KB)
Abstract:

In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.

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