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Mathematical Control & Related Fields

September & December 2018 , Volume 8 , Issue 3&4

Special issue dedicated to Jiongmin Yong on the occasion of his 60th birthday

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Preface: A tribute to professor Jiongmin Yong on his 60th birthday
Hongwei Lou, Qi Lü, Gengsheng Wang and Xu Zhang
2018, 8(3&4): ⅰ-ⅰ doi: 10.3934/mcrf.201803i +[Abstract](372) +[HTML](213) +[PDF](77.17KB)
Abstract:
Jiongmin Yong's mathematical works in recent thirty years
Shuping Chen
2018, 8(3&4): 491-500 doi: 10.3934/mcrf.2018048 +[Abstract](331) +[HTML](192) +[PDF](333.46KB)
Abstract:
Existence for nonlinear finite dimensional stochastic differential equations of subgradient type
Viorel Barbu
2018, 8(3&4): 501-508 doi: 10.3934/mcrf.2018020 +[Abstract](272) +[HTML](226) +[PDF](342.93KB)
Abstract:

One proves via variational techniques the existence and uniqueness of a strong solution to the stochastic differential equation \begin{document}$dX+{\partial} {\varphi} (t,X)dt\ni \sum\limits^N_{i = 1}σ_i(X)d{β}_i,\ X(0) = x,$\end{document} where \begin{document}${\partial}{\varphi} :{\mathbb{R}}^d\to2^{{\mathbb{R}}^d}$\end{document} is the subdifferential of a convex function \begin{document}${\varphi}:{\mathbb{R}}^d\to{\mathbb{R}}$\end{document} and \begin{document}$σ_i∈ L({\mathbb{R}}^d,{\mathbb{R}}^d)$\end{document}, \begin{document}$1≤ d<{∞}$\end{document}.

Value function for regional control problems via dynamic programming and Pontryagin maximum principle
Guy Barles, Ariela Briani and Emmanuel Trélat
2018, 8(3&4): 509-533 doi: 10.3934/mcrf.2018021 +[Abstract](348) +[HTML](171) +[PDF](502.79KB)
Abstract:

In this paper we consider regional deterministic finite-dimensional optimal control problems, where the dynamics and the cost functional depend on the region of the state space where one is and have discontinuities at their interface.

Under the assumption that optimal trajectories have a locally finite number of switchings (i.e., no Zeno phenomenon), we use the duplication technique to show that the value function of the regional optimal control problem is the minimum over all possible structures of trajectories of value functions associated with classical optimal control problems settled over fixed structures, each of them being the restriction to some submanifold of the value function of a classical optimal control problem in higher dimension.

The lifting duplication technique is thus seen as a kind of desingularization of the value function of the regional optimal control problem.

In turn, we establish sensitivity relations for regional optimal control problems and we prove that the regularity of the value function of such problems is the same (i.e., is not more degenerate) than the one of the higher-dimensional classical optimal control problem that lifts the problem.

Necessary conditions for infinite horizon optimal control problems with state constraints
Vincenzo Basco, Piermarco Cannarsa and Hélène Frankowska
2018, 8(3&4): 535-555 doi: 10.3934/mcrf.2018022 +[Abstract](328) +[HTML](162) +[PDF](520.42KB)
Abstract:

Partial and full sensitivity relations are obtained for nonauto-nomous optimal control problems with infinite horizon subject to state constraints, assuming the associated value function to be locally Lipschitz in the state. Sufficient structural conditions are given to ensure such a Lipschitz regularity in presence of a positive discount factor, as it is typical of macroeconomics models.

Numerical study of polynomial feedback laws for a bilinear control problem
Tobias Breiten, Karl Kunisch and Laurent Pfeiffer
2018, 8(3&4): 557-582 doi: 10.3934/mcrf.2018023 +[Abstract](296) +[HTML](174) +[PDF](1428.79KB)
Abstract:

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

General boundary value problems of the Korteweg-de Vries equation on a bounded domain
Roberto A. Capistrano-Filho, Shuming Sun and Bing-Yu Zhang
2018, 8(3&4): 583-605 doi: 10.3934/mcrf.2018024 +[Abstract](336) +[HTML](157) +[PDF](493.23KB)
Abstract:

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval

subject to the nonhomogeneous boundary conditions,

where

and \begin{document}$a_{ij}, \ b_{ij}$\end{document} (\begin{document}$j = 0, 1, 2$\end{document} and \begin{document}$ i = 1, 2, 3$\end{document}) are real constants. Under some general assumptions imposed on the coefficients \begin{document}$a_{ij}, \ b_{ij}$\end{document}, the IBVP (0.1)-(0.2) is shown to be locally well-posed in the space \begin{document}$H^s (0, L)$\end{document} for any \begin{document}$s \ge 0$\end{document} with \begin{document}$φ ∈ H^s (0, L)$\end{document} and boundary values \begin{document}$h_j$\end{document}, \begin{document}$j = 1, 2, 3$\end{document}, belonging to some appropriate spaces with optimal regularity.

Analysis and optimal control of some quasilinear parabolic equations
Eduardo Casas and Konstantinos Chrysafinos
2018, 8(3&4): 607-623 doi: 10.3934/mcrf.2018025 +[Abstract](302) +[HTML](193) +[PDF](432.15KB)
Abstract:

In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

Recovery of local volatility for financial assets with mean-reverting price processes
Qihong Chen
2018, 8(3&4): 625-635 doi: 10.3934/mcrf.2018026 +[Abstract](336) +[HTML](196) +[PDF](308.6KB)
Abstract:

This article is concerned with the model calibration for financial assets with mean-reverting price processes, which is an important topic in mathematical finance.

The discussion focuses on the recovery of local volatility from market data for Schwartz(1997) model. It is formulated as an inverse parabolic problem, and the necessary condition for determining the local volatility is derived under the optimal control framework. An iterative algorithm is provided to solve the optimality system and a synthetic numerical example is provided to illustrate the effectiveness.

Weak laws of large numbers for sublinear expectation
Zengjing Chen, Qingyang Liu and Gaofeng Zong
2018, 8(3&4): 637-651 doi: 10.3934/mcrf.2018027 +[Abstract](253) +[HTML](180) +[PDF](394.0KB)
Abstract:

In this paper we study the weak laws of large numbers for sublinear expectation. We prove that, without any moment condition, the weak laws of large numbers hold in the sense of convergence in capacity induced by some general sublinear expectations. For some specific sublinear expectation, for instance, mean deviation functional and one-side moment coherent risk measure, we also give weak laws of large numbers for corresponding capacity.

Linear quadratic mean-field-game of backward stochastic differential systems
Kai Du, Jianhui Huang and Zhen Wu
2018, 8(3&4): 653-678 doi: 10.3934/mcrf.2018028 +[Abstract](328) +[HTML](285) +[PDF](514.51KB)
Abstract:

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Inverse S-shaped probability weighting and its impact on investment
Xue Dong He, Roy Kouwenberg and Xun Yu Zhou
2018, 8(3&4): 679-706 doi: 10.3934/mcrf.2018029 +[Abstract](351) +[HTML](201) +[PDF](556.27KB)
Abstract:

In this paper we analyze how changes in inverse S-shaped probability weighting influence optimal portfolio choice in a rank-dependent utility model. We derive sufficient conditions for the existence of an optimal solution of the investment problem, and then define the notion of a more inverse S-shaped probability weighting function. We show that an increase in inverse S-shaped weighting typically leads to a lower allocation to the risky asset, regardless of whether the return distribution is skewed left or right, as long as it offers a non-negligible risk premium. Only for lottery stocks with poor expected returns and extremely positive skewness does an increase in inverse S-shaped probability weighting lead to larger portfolio allocations.

Null controllability of the Lotka-McKendrick system with spatial diffusion
Nicolas Hegoburu and Marius Tucsnak
2018, 8(3&4): 707-720 doi: 10.3934/mcrf.2018030 +[Abstract](262) +[HTML](174) +[PDF](488.73KB)
Abstract:

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

Quadratic BSDEs with mean reflection
Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo and Falei Wang
2018, 8(3&4): 721-738 doi: 10.3934/mcrf.2018031 +[Abstract](302) +[HTML](185) +[PDF](512.65KB)
Abstract:

The present paper is devoted to the study of the well-posedness of BSDEs with mean reflection whenever the generator has quadratic growth in the \begin{document}$z$\end{document} argument. This work is the sequel of [6] in which a notion of BSDEs with mean reflection is developed to tackle the super-hedging problem under running risk management constraints. By the contraction mapping argument, we first prove that the quadratic BSDE with mean reflection admits a unique deterministic flat local solution on a small time interval whenever the terminal value is bounded. Moreover, we build the global solution on the whole time interval by stitching local solutions when the generator is uniformly bounded with respect to the \begin{document}$y$\end{document} argument.

Nonlinear backward stochastic evolutionary equations driven by a space-time white noise
Ying Hu and Shanjian Tang
2018, 8(3&4): 739-751 doi: 10.3934/mcrf.2018032 +[Abstract](230) +[HTML](153) +[PDF](343.05KB)
Abstract:

We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and then give an existence and uniqueness result for nonlinear backward stochastic evolutionary equations. A dual argument plays a crucial role in the proof of these results. Finally, an example is given to illustrate the existence and uniqueness result.

Recursive utility optimization with concave coefficients
Shaolin Ji and Xiaomin Shi
2018, 8(3&4): 753-775 doi: 10.3934/mcrf.2018033 +[Abstract](315) +[HTML](164) +[PDF](509.56KB)
Abstract:

This paper concerns the recursive utility maximization problem. We assume that the coefficients of the wealth equation and the recursive utility are concave. Then some interesting and important cases with nonlinear and nonsmooth coefficients satisfy our assumption. After given an equivalent backward formulation of our problem, we employ the Fenchel-Legendre transform and derive the corresponding variational formulation. By the convex duality method, the primal "sup-inf" problem is translated to a dual minimization problem and the saddle point of our problem is derived. Finally, we obtain the optimal terminal wealth. To illustrate our results, three cases for investors with ambiguity aversion are explicitly worked out under some special assumptions.

Feedback stabilization with one simultaneous control for systems of parabolic equations
Cătălin-George Lefter and Elena-Alexandra Melnig
2018, 8(3&4): 777-787 doi: 10.3934/mcrf.2018034 +[Abstract](247) +[HTML](185) +[PDF](346.33KB)
Abstract:

In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.

Weak stability of a laminated beam
Yanfang Li, Zhuangyi Liu and Yang Wang
2018, 8(3&4): 789-808 doi: 10.3934/mcrf.2018035 +[Abstract](225) +[HTML](209) +[PDF](415.96KB)
Abstract:

In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.

Optimal control problems for some ordinary differential equations with behavior of blowup or quenching
Ping Lin and Weihan Wang
2018, 8(3&4): 809-828 doi: 10.3934/mcrf.2018036 +[Abstract](279) +[HTML](177) +[PDF](453.37KB)
Abstract:

This paper is concerned with some optimal control problems for equations with blowup or quenching property. We first study the existence and Pontryagin's maximum principle for optimal controls which have the minimal energy among all the controls whose corresponding solutions blow up at the right-hand time end-point of a given functional. Then, the same problem for quenching case is discussed. Finally, we establish Pontryagin's maximum principle for optimal controls of extended problems after quenching.

Controllability and observability of some coupled stochastic parabolic systems
Lingyang Liu and Xu Liu
2018, 8(3&4): 829-854 doi: 10.3934/mcrf.2018037 +[Abstract](225) +[HTML](218) +[PDF](482.1KB)
Abstract:

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.

Minimization of the elliptic higher eigenvalues for multiphase anisotropic conductors
Hongwei Lou and Xueyuan Yin
2018, 8(3&4): 855-877 doi: 10.3934/mcrf.2018038 +[Abstract](224) +[HTML](190) +[PDF](447.42KB)
Abstract:

Higher eigenvalues of composite materials for anisotropic conductors are considered. To get the existence result for minimizing problems, relaxed problems are introduced by the homogenization method. Then, necessary conditions for minimizers are yielded. Based on the necessary conditions, it is shown that in some cases, optimal conductivities of relaxed minimizing problems can be replaced equivalently by a weighted harmonic mean of conductivities.

Recurrence for switching diffusion with past dependent switching and countable state space
Dang H. Nguyen and George Yin
2018, 8(3&4): 879-897 doi: 10.3934/mcrf.2018039 +[Abstract](230) +[HTML](160) +[PDF](437.21KB)
Abstract:

This work continues and substantially extends our recent work on switching diffusions with the switching processes that depend on the past states and that take values in a countable state space. That is, the discrete component of the two-component process takes values in a countably infinite set and its switching rate at current time depends on the value of the continuous component involving past history. This paper focuses on recurrence, positive recurrence, and weak stabilization of such systems. In particular, the paper aims to providing more verifiable conditions on recurrence and positive recurrence and related issues. Assuming that the system is linearizable, it provides feasible conditions focusing on the coefficients of the systems for positive recurrence. Then linear feedback controls for weak stabilization are considered. Some illustrative examples are also given.

Carleman commutator approach in logarithmic convexity for parabolic equations
Kim Dang Phung
2018, 8(3&4): 899-933 doi: 10.3934/mcrf.2018040 +[Abstract](247) +[HTML](191) +[PDF](595.81KB)
Abstract:

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

Controllability under positivity constraints of semilinear heat equations
Dario Pighin and Enrique Zuazua
2018, 8(3&4): 935-964 doi: 10.3934/mcrf.2018041 +[Abstract](306) +[HTML](158) +[PDF](795.11KB)
Abstract:

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed.

In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive.

More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory.

We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.

Switching between a pair of stocks: An optimal trading rule
Jingzhi Tie and Qing Zhang
2018, 8(3&4): 965-999 doi: 10.3934/mcrf.2018042 +[Abstract](356) +[HTML](199) +[PDF](1490.05KB)
Abstract:

This paper is about a stock trading rule involving two stocks. The trader may have a long position in either stock or in cash. She may also switch between them any time. Her objective is to trade over time to maximize an expected return. In this paper, we reduce the problem to the optimal trading control problem under a geometric Brownian motion model with regime switching. We use a two-state Markov chain to capture the general market modes. In particular, a single market cycle consisting of a bull market followed by a bear market is considered. We also impose a fixed percentage cost on each transaction. We focus on simple threshold-type policies and study all possible combinations. We establish algebraic equations to characterize these threshold levels. We also present sufficient conditions that guarantee the optimality of these policies. Finally, some numerical examples are provided to illustrate our results.

Admissible controls and controllable sets for a linear time-varying ordinary differential equation
Lijuan Wang and Yashan Xu
2018, 8(3&4): 1001-1019 doi: 10.3934/mcrf.2018043 +[Abstract](253) +[HTML](158) +[PDF](502.32KB)
Abstract:

In this paper, for a time optimal control problem governed by a linear time-varying ordinary differential equation, we give a description to check whether the set of admissible controls is nonempty or not by finite times.

Forward backward SDEs in weak formulation
Haiyang Wang and Jianfeng Zhang
2018, 8(3&4): 1021-1049 doi: 10.3934/mcrf.2018044 +[Abstract](226) +[HTML](150) +[PDF](583.34KB)
Abstract:

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.

Time-inconsistent recursive zero-sum stochastic differential games
Qingmeng Wei and Zhiyong Yu
2018, 8(3&4): 1051-1079 doi: 10.3934/mcrf.2018045 +[Abstract](227) +[HTML](165) +[PDF](556.58KB)
Abstract:

In this paper, a kind of time-inconsistent recursive zero-sum stochastic differential game problems are studied by a hierarchical backward sequence of time-consistent subgames. The notion of feedback control-strategy law is adopted to constitute a closed-loop formulation. Instead of the time-inconsistent saddle points, a new concept named equilibrium saddle points is introduced and investigated, which is time-consistent and can be regarded as a local approximate saddle point in a proper sense. Moreover, a couple of equilibrium Hamilton-Jacobi-Bellman-Isaacs equations are obtained to characterize the equilibrium values and construct the equilibrium saddle points.

Optimal actuator location of the minimum norm controls for stochastic heat equations
Donghui Yang and Jie Zhong
2018, 8(3&4): 1081-1095 doi: 10.3934/mcrf.2018046 +[Abstract](214) +[HTML](160) +[PDF](387.07KB)
Abstract:

In this paper, we study the approximate null controllability for the stochastic heat equation with the control acting on a measurable subset, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.

Quantitative unique continuation for the heat equation with Coulomb potentials
Can Zhang
2018, 8(3&4): 1097-1116 doi: 10.3934/mcrf.2018047 +[Abstract](241) +[HTML](168) +[PDF](459.95KB)
Abstract:

In this paper, we establish a Hölder-type quantitative estimate of unique continuation for solutions to the heat equation with Coulomb potentials in either a bounded convex domain or a \begin{document}$C^2$\end{document}-smooth bounded domain. The approach is based on the frequency function method, as well as some parabolic-type Hardy inequalities.

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