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Mathematical Control and Related Fields

June 2022 , Volume 12 , Issue 2

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Optimal control of perfect plasticity part I: Stress tracking
Christian Meyer and Stephan Walther
2022, 12(2): 275-301 doi: 10.3934/mcrf.2021022 +[Abstract](1219) +[HTML](420) +[PDF](1311.99KB)
Abstract:

The paper is concerned with an optimal control problem governed by the rate-independent system of quasi-static perfect elasto-plasticity. The objective is to optimize the stress field by controlling the displacement at prescribed parts of the boundary. The control thus enters the system in the Dirichlet boundary conditions. Therefore, the safe load condition is automatically fulfilled so that the system admits a solution, whose stress field is unique. This gives rise to a well defined control-to-state operator, which is continuous but not Gâteaux differentiable. The control-to-state map is therefore regularized, first by means of the Yosida regularization resp. viscous approximation and then by a second smoothing in order to obtain a smooth problem. The approximation of global minimizers of the original non-smooth optimal control problem is shown and optimality conditions for the regularized problem are established. A numerical example illustrates the feasibility of the smoothing approach.

Solvable approximations of 3-dimensional almost-Riemannian structures
Philippe Jouan and Ronald Manríquez
2022, 12(2): 303-326 doi: 10.3934/mcrf.2021023 +[Abstract](1170) +[HTML](435) +[PDF](702.96KB)
Abstract:

In some cases, the nilpotent approximation of an almost-Riemannian structure can degenerate into a constant rank sub-Riemannian one. In those cases, the nilpotent approximation can be replaced by a solvable one that turns out to be a linear ARS on a nilpotent Lie group or a homogeneous space.

The distance defined by the solvable approximation is analyzed in the 3D-generic cases. It is shown that it is a better approximation of the original distance than the nilpotent one.

Controllability under positive constraints for quasilinear parabolic PDEs
Miguel R. Nuñez-Chávez
2022, 12(2): 327-341 doi: 10.3934/mcrf.2021024 +[Abstract](1172) +[HTML](415) +[PDF](446.87KB)
Abstract:

This paper deals with the analysis of the internal controllability with constraint of positive kind of a quasilinear parabolic PDE. We prove two results about this PDE: First, we prove a global steady state constrained controllability result. For this purpose, we employ the called "stair-case method". And second, we prove a global trajectory constrained controllability result. For this purpose, we employ the well-known "stabilization property" in \begin{document}$ L^2 $\end{document} norms. Furthermore, for both results an important argument is needed: the exact local controllability to trajectories. Then we prove the positivity of the minimal controllability time using arguments of comparison principle. Some additional comments and open problems concerning other systems are presented.

A nonzero-sum risk-sensitive stochastic differential game in the orthant
Mrinal K. Ghosh and Somnath Pradhan
2022, 12(2): 343-370 doi: 10.3934/mcrf.2021025 +[Abstract](1127) +[HTML](435) +[PDF](586.79KB)
Abstract:

We study a nonzero-sum risk-sensitive stochastic differential game for controlled reflecting diffusion processes in the nonnegative orthant. We treat two cost evaluation criteria, namely, discounted cost and ergodic cost. Under certain assumptions, we establish the existence of Nash equilibria. Also, we completely characterize a Nash equilibrium for the ergodic cost criterion in the space of stationary Markov strategies.

Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems
Jun Moon
2022, 12(2): 371-404 doi: 10.3934/mcrf.2021026 +[Abstract](1606) +[HTML](410) +[PDF](676.03KB)
Abstract:

In this paper, we consider linear-quadratic (LQ) leader-follower Stackelberg differential games for mean-field type stochastic systems with jump diffusions, where the system includes mean-field variables, i.e., the expected value of state and control variables. We first solve the LQ mean-field type control problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the coupled integro-Riccati differential equations (CIRDEs) via the Four-Step Scheme. Next, we solve the problem of the leader, which is the LQ control problem subject to the mean-field type forward-backward stochastic system with jump diffusions, where the constraint characterizes the rational behavior of the follower. Using the variational approach, we obtain the (mean-field type) stochastic maximum principle. However, to obtain the state-feedback representation of the open-loop optimal solution of the leader, there is a technical challenge due to the jump process. We consider two different cases, in which the state-feedback type control in terms of the CIRDEs can be characterized by generalizing the Four-Step Scheme. We finally show that the state-feedback type controls of the open-loop optimal solutions for the leader and the follower constitute the Stackelberg equilibrium.

Feedback stabilization of parabolic systems with input delay
Imene Aicha Djebour, Takéo Takahashi and Julie Valein
2022, 12(2): 405-420 doi: 10.3934/mcrf.2021027 +[Abstract](1084) +[HTML](412) +[PDF](516.79KB)
Abstract:

This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the \begin{document}$ N $\end{document}-dimensional linear reaction-convection-diffusion equation with \begin{document}$ N\geq 1 $\end{document} and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.

On the nonuniqueness and instability of solutions of tracking-type optimal control problems
Constantin Christof and Dominik Hafemeyer
2022, 12(2): 421-431 doi: 10.3934/mcrf.2021028 +[Abstract](1249) +[HTML](374) +[PDF](438.98KB)
Abstract:

We study tracking-type optimal control problems that involve a non-affine, weak-to-weak continuous control-to-state mapping, a desired state \begin{document}$ y_d $\end{document}, and a desired control \begin{document}$ u_d $\end{document}. It is proved that such problems are always nonuniquely solvable for certain choices of the tuple \begin{document}$ (y_d, u_d) $\end{document} and instable in the sense that the set of solutions (interpreted as a multivalued function of \begin{document}$ (y_d, u_d) $\end{document}) does not admit a continuous selection.

Local Kalman rank condition for linear time varying systems
Hamid Maarouf
2022, 12(2): 433-446 doi: 10.3934/mcrf.2021029 +[Abstract](1327) +[HTML](451) +[PDF](396.73KB)
Abstract:

In this paper, we study some non-negative integers related to a linear time varying system and to some Krylov sub-spaces associated to this system. Such integers are similar to the controllability indices and have been used in the literature to derive results on the controllability of linear systems. The purpose of this paper goes in the same direction by studying the local behavior of these integers especially nearby instants in the time interval with some maximal rank condition and then apply them to get some results which generalize the mentioned existing results.

Local well-posedness for a class of 1D Boussinesq systems
Alex M. Montes and Ricardo Córdoba
2022, 12(2): 447-473 doi: 10.3934/mcrf.2021030 +[Abstract](1044) +[HTML](415) +[PDF](456.95KB)
Abstract:

In this paper we study the local well-posedness for the Cauchy problem associated with a special class of one-dimensional Boussinesq systems that model the evolution of long water waves with small amplitude in the presence of surface tension.

Maximum principle for discrete-time stochastic optimal control problem and stochastic game
Zhen Wu and Feng Zhang
2022, 12(2): 475-493 doi: 10.3934/mcrf.2021031 +[Abstract](1194) +[HTML](355) +[PDF](609.58KB)
Abstract:

This paper is first concerned with one kind of discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. The results are then extended to two kinds of discrete-time stochastic games. Two illustrative examples are studied, for which the explicit optimal strategies are given. This paper establishes a rigorous version of discrete-time stochastic maximum principle in a clear and concise way and paves a road for further related topics.

Existence and cost of boundary controls for a degenerate/singular parabolic equation
U. Biccari, V. Hernández-Santamaría and J. Vancostenoble
2022, 12(2): 495-530 doi: 10.3934/mcrf.2021032 +[Abstract](1011) +[HTML](351) +[PDF](696.54KB)
Abstract:

In this paper, we consider the following degenerate/singular parabolic equation

where \begin{document}$ 0\leq \alpha <1 $\end{document} and \begin{document}$ \mu\leq (1-\alpha)^2/4 $\end{document} are two real parameters. We prove the boundary null controllability by means of a \begin{document}$ H^1(0,T) $\end{document} control acting either at \begin{document}$ x = 1 $\end{document} or at the point of degeneracy and singularity \begin{document}$ x = 0 $\end{document}. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters \begin{document}$ \alpha $\end{document} and \begin{document}$ \mu $\end{document}. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.

Optimal investment and reinsurance of insurers with lognormal stochastic factor model
Hiroaki Hata and Li-Hsien Sun
2022, 12(2): 531-566 doi: 10.3934/mcrf.2021033 +[Abstract](1109) +[HTML](387) +[PDF](798.05KB)
Abstract:

We propose the stochastic factor model of optimal investment and reinsurance of insurers where the wealth processes are described by a bank account and a risk asset for investment and a Cramér-Lundberg process for reinsurance. The optimization is obtained through maximizing the exponential utility. Owing to the claims driven by a Poisson process, the proposed optimization problem is naturally treated as a jump-diffusion control problem. Applying the dynamic programming, we have the Hamilton-Jacobi-Bellman (HJB) equations and the corresponding explicit solution for the corresponding HJB. Hence, the optimal values and optimal strategies can be obtained. Finally, in numerical analysis, we illustrate the performance of the proposed optimization according to the results of the corresponding value function. In addition, compared to the wealth process without investment, the efficiency of the proposed optimization is discussed in terms of ruin probabilities.

2021 Impact Factor: 1.141
5 Year Impact Factor: 1.362
2021 CiteScore: 2.4

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