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Robust stochastic control modeling of dam discharge to suppress overgrowth of downstream harmful algae

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY 2018年第3期34卷 338-354页

作者： Yoshioka, Hidekazu Yaegashi, Yuta Shimane Univ Fac Life & Environm Sci Matsue Shimane 6908504 Japan Kyoto Univ Grad Sch Agr Kyoto Japan Japan Soc Promot Sci Kyoto Japan

The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal of the problem is to find a fit-for-purpose and environmentally sound operation policy of the flow discharge from a dam so that overgrowth of the harmful algae Cladophora glomerataKutzing in its downstream river is effectively suppressed. A minimal stochastic differential equation for the algae growth dynamics with uncertain growth rate is first presented. The performance index to be maximized by the operator of the dam while minimized by nature is formulated within the framework of differential games. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs equation whose solution determines the worst-case optimal operation policy of the dam, ie, the policy that the operator wants to find. Application of the model to overgrowth suppression of Cladophora glomerataKutzing just downstream of a dam in a Japanese river is then carried out. Values of the model parameters are identified with which the model successfully reproduces the observed population dynamics. A series of numerical experiments are performed to find the most effective operation policy of the dam based on a relaxation of the current policy.

关键词： Cladophora glomerata Kutzing dynamic programming principle flow discharge Hamilton-Jacobi-Bellman-Isaacs equation model misspecification multiplier robust control

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Optimal stopping with random maturity under nonlinear expectations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2017年第8期127卷 2586-2629页

作者： Bayraktar, Erhan Yao, Song Univ Michigan Dept Math Ann Arbor MI 48109 USA Univ Pittsburgh Dept Math Pittsburgh PA 15260 USA

We analyze an optimal stopping problem sup(gamma is an element of T) (xi) over bar 0[y(gamma Lambda tau 0)] with random maturity to under a nonlinear expectation (xi) over bar0[.] := sup(P is an element of P) Eg[.], where P is a weakly compact set of mutually singular probabilities. The maturity tau(0) is specified as the hitting time to level 0 of some continuous index process X at which the payoff process g is even allowed to have a positive jump. When P collects a variety of semimartingale measures, the optimal stopping problem can be viewed as a discretionary stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow. We utilize a martingale approach to construct an optimal pair (P-*, y(*)) for sup((P, gamma)is an element of P X T) Ep[y(gamma Lambda tau 0)], in which y(*) is the first time y meets the limit. L of its approximating (xi) over bar -Snell envelopes. To overcome the technical subtleties caused by the mutual singularity of probabilities in P and the discontinuity of the payoff process y, we approximate tau(0) by an increasing sequence of Lipschitz continuous stopping times and approximate y by a sequence of uniformly continuous processes. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Discretionary stopping Random maturity Controls in weak formulation Optimal stopping Nonlinear expectation Weak stability under pasting Lipschitz continuous stopping time dynamic programming principle Martingale approach

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BSDES IN GAMES,COUPLED WITH THE VALUE *** NONLOCAL BELLMAN-ISAACS EQUATIONS

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Acta Mathematica Scientia 2017年第5期37卷 1497-1518页

作者：郝涛李娟 School of Statistics Shandong University of Finance and Economics School of Mathematics and Statistics Shandong University

We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.

关键词： Mc Kean-Vlasov SDE BSDE coupled with the lower and the upper value functions dynamic programming principle mean-field BSDE viscosity solution coupled nonlocal HJB-Isaacs equation Isaacs' condition

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A uniform Tauberian theorem in dynamic games

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SBORNIK MATHEMATICS 2018年第1期209卷 122-144页

作者： Khlopin, Dmitrii V. Russian Acad Sci Krasovskii Inst Math & Mech Ural Branch Ekaterinburg Russia

Antagonistic dynamic games including games represented in normal form are considered. The asymptotic behaviour of value in these games is investigated as the game horizon tends to infinity (Cesaro mean) and as the discounting parameter tends to zero (Abel mean). The corresponding Abelian-Tauberian theorem is established: it is demonstrated that in both families the game value uniformly converges to the same limit, provided that at least one of the limits exists. Analogues of one-sided Tauberian theorems are obtained. An example shows that the requirements are essential even for control problems.

关键词： dynamic programming principle games with a saddle point Tauberian theorem

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OPTIMAL CONTROL OF BRANCHING DIFFUSION PROCESSES: A FINITE HORIZON PROBLEM

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ANNALS OF APPLIED PROBABILITY 2018年第1期28卷 1-34页

作者： Claisse, Julien Ecole Polytech Ctr Math Appl F-91128 Palaiseau France

In this paper, we aim to develop the stochastic control theory of branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio [J. Math. Kyoto Univ. 25 (1985) 549-575] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to derive a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.

关键词： Stochastic control branching diffusion process dynamic programming principle Hamilton-Jacobi-Bellman equation viscosity solution

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Towards Parallelizable Sampling–based Nonlinear Model Predictive Control

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IFAC-PapersOnLine 2017年第1期50卷 13176-13181页

作者： Bobiti, R.V. Lazar, M. Department of Electrical Engineering Eindhoven University of Technology Netherlands

This paper proposes a new sampling–based nonlinear model predictive control (MPC) algorithm, with a bound on complexity quadratic in the prediction horizon N and linear in the number of samples. The idea of the proposed algorithm is to use the sequence of predicted inputs from the previous time step as a warm start, and to iteratively update this sequence by changing its elements one by one, starting from the last predicted input and ending with the first predicted input. This strategy, which resembles the dynamic programming principle, allows for parallelization up to a certain level and yields a suboptimal nonlinear MPC algorithm with guaranteed recursive feasibility, stability and improved cost function at every iteration, which is suitable for real–time implementation. The complexity of the algorithm per each time step in the prediction horizon depends only on the horizon, the number of samples and parallel threads, and it is independent of the measured system state. Comparisons with the fmincon nonlinear optimization solver on a benchmark example indicates that, as the simulation time progresses, the proposed algorithm converges rapidly to the “optimal” solution, even when using a small number of samples. © 2017

关键词： dynamic programming Computational complexity Constrained optimization Cost functions Iterative methods Model predictive control Nonlinear programming Nonlinear systems Predictive control systems Constrained controls dynamic programming principle Embedded model predictive control Nonlinear model predictive control Nonlinear optimization solver Nonlinear predictive control Number of samples Prediction horizon

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Linear-quadratic optimal sampled-data control problems: Convergence result and Riccati theory

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AUTOMATICA 2017年 79卷 273-281页

作者： Bourdin, Loic Trelat, Emmanuel Univ Limoges Inst Rech XLIM Pole Math CNRS UMR 7252 Limoges France UPMC Univ Paris 06 Sorbonne Univ CNRS UMR 7598 Lab Jacques Louis Lions F-75005 Paris France

We consider linear-quadratic optimal sampled-data control problems, where the state evolves continuously in time according to a linear control system and the control is sampled, i.e., is piecewise constant over a subdivision of the time interval, and the cost is quadratic. As a first result, we prove that, as the sampling periods tend to zero, the optimal sampled-data controls converge pointwise to the optimal permanent control. Then, we extend the classical Riccati theory to the sampled-data control framework, by developing two different approaches: the first one uses a recently established version of the Pontryagin maximum principle for optimal sampled-data control problems, and the second one uses an adequate version of the dynamic programming principle. In turn, we obtain a closed-loop expression for optimal sampled-data controls of linear-quadratic problems. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Optimal control Linear-quadratic problems Sampled-data control Digital control Convergence Pontryagin maximum principle Riccati theory dynamic programming principle

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Optimal consumption-investment strategy under the Vasicek model: HARA utility and Legendre transform

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INSURANCE MATHEMATICS & ECONOMICS 2017年 72卷 215-227页

作者： Chang, Hao Chang, Kai Tianjin Polytech Univ Sch Sci Tianjin 300387 Peoples R China Tianjin Univ Coll Management & Econ Tianjin 300072 Peoples R China Zhejiang Univ Finance & Econ Sch Finance Hangzhou 310018 Zhejiang Peoples R China

This paper studies the optimal consumption-investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption-investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption-investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Consumption-investment problem The Vasicek model HARA utility dynamic programming principle Legendre transform Closed-form solution

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Stochastic modeling and control of bioreactors

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IFAC-PapersOnLine 2017年第1期50卷 12611-12616页

作者： Fontbona, J. Ramírez C., H. Riquelme, V. Silva, F.J. Universidad de Chile Beauchef 851 Casilla 170-3 Santiago 3 Chile Institut de recherche XLIM-DMI UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges Limoges87060 France

In this work we propose a stochastic model for a sequencing-batch reactor (SBR) and for a chemostat. Both models are described by systems of Stochastic Differential Equations (SDEs), which are obtained as limits of suitable Markov Processes characterizing the microscopic behavior. We study the existence of solutions of the obtained equations as well as some properties, among which the possible extinction of the biomass is the most remarkable feature. The implications of this behavior are illustrated in the problem consisting in maximizing the probability of reaching a desired depollution level prior to biomass extinction. © 2017

关键词： Stochastic models Batch reactors Chemostats Differential equations dynamic programming Markov processes Pollution control Stochastic control systems Stochastic systems Demographic stochasticity Depollution dynamic programming principle Existence of Solutions Microscopic behavior Sequencing batch reactors Stochastic control Systems of stochastic differential equations

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MAXIMAL OPERATORS FOR THE p-LAPLACIAN FAMILY

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PACIFIC JOURNAL OF MATHEMATICS 2017年第2期287卷 257-295页

作者： Blanc, Pablo Pinasco, Juan P. Rossi, Julio D. Univ Buenos Aires FCEyN Dept Matemat Ciudad UnivPabellon 1 RA-1428 Buenos Aires DF Argentina

We prove existence and uniqueness of viscosity solutions for the problem max {-Delta(p1)u(x), -Delta(p2)u(x)} = f(x) in a bounded smooth domain Omega subset of R-N with u = g on partial derivative Omega. Here -Delta(p)u = (N + p)(-1) vertical bar Du vertical bar(2-p) div (vertical bar Du vertical bar(p-2)Du) is the 1-homogeneous p-Laplacian and we assume that 2 <= p(1), p(2) <= infinity. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-of-war game (without noise) or playing at random. Moreover, the operator max {-Delta(p1)u(x), -Delta(p2)u(x)} provides a natural analogue with respect to p-Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory.

关键词： Dirichlet boundary conditions dynamic programming principle p-Laplacian tug-of-war games

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