Finite-horizon Optimal Problem Research Articles

Portfolio Problem for the α−Hypergeometric Stochastic Volatility Model with Consumption

This paper solves the finite horizon optimal consumption-investment problem for an investor that trades in the α−hypergeometric stochastic volatility model, with preferences based on a power utility function. To find the optimal strategy, we follow the dynamic programming approach. First, we perform a suitable change of variables in order to fully transform the non-linear Hamilton-Jacobi-Bellman (HJB) equation into a semilinear one. Next, we apply the Girsanov theorem to deduce an implicit Feynman-Kac formula depending upon the volatility process. The operator defined by the Feynman-Kac representation is shown to be a contraction operator on a designed function space. Finally, the Banach fixed point theorem yields the existence of a solution to the HJB equation in the designed space. Moreover, we apply a verification theorem to guarantee that the solution of the HJB equation coincides with the value function.

An Explicit Dual Control Approach for Constrained Reference Tracking of Uncertain Linear Systems

A finite horizon optimal tracking problem is considered for linear dynamical systems subject to parametric uncertainties in the state-space matrices and exogenous disturbances. A suboptimal solution is proposed using a model predictive control (MPC) based explicit dual control approach which enables active uncertainty learning. A novel algorithm for the design of robustly invariant online terminal sets and terminal controllers is presented. Set membership identification is used to update the parameter uncertainty online. A predicted worst-case cost is used in the MPC optimization problem to model the dual effect of the control input. The cost-to-go is estimated using contractivity of the proposed terminal set and the remaining time horizon, so that the optimizer can estimate future benefits of exploration. The proposed dual control algorithm ensures robust constraint satisfaction and recursive feasibility, and navigates the exploration-exploitation trade-off using a robust performance metric.

Modeling and Finite-Horizon MPC for a Boiler-Turbine System Using Minimal Realization State-Space Model

This paper aims to address a finite-horizon model predictive control (MPC) for non-linear drum-type boiler-turbine system using a system-identification method. Considering that the strong state coupling of a non-linear mechanism model, the subspace identification method is first utilized to obtain a linear state-space model, and transformed into an input–output model. By taking the inputs and outputs of the input–output model as system states, an augmented non-minimal state-space (NMSS) model of state measurable is constructed. In order to reduce the computation burden, the augmented NMSS model is further transformed into a canonical formulation by adopting a Kalman decomposition. Based on the minimal realization state-space model, the MPC controller is parameterized as a finite-horizon optimization problem. Finally, simulations are performed and evaluated the performance of the proposed method, and the simulation results show that: the linear model approximate the non-linear system accurately; the proposed MPC method can achieve a satisfactory stable control performance; and the computation time 18.388 s for the overall optimization problem also illustrates the real-time performance effectively.

Stabilization of Discrete-Time Multiplicative-Noise System Under Decentralized Controllers

In this article, we study the mean-square stabilization of discrete-time multiplicative-noise stochastic systems under decentralized controllers. Two controllers are involved in the system where each controller has access to different information and one information set is contained by another one. The adopted information structure is adapted open-loop. The main contribution of the article is twofold. On one hand, we give the unique explicit solution of the finite-horizon optimization problem in terms of difference Riccati equations. On the other hand, we characterize the equivalent condition for the mean-square stabilization under the decentralized controllers via the corresponding algebraic Riccati equations. Moreover, we derive the optimal and stabilizing solution for the infinite-horizon optimization problem.

Solution existence theorems for finite horizon optimal economic growth problems

The solution existence of finite horizon optimal economic growth problems is studied by invoking Filippov's Existence Theorem for optimal control problems from the monograph of Cesari L. [Optimization theory and applications. New York: Springer-Verlag; 1983]. Our results are obtained not only for general problems but also for typical ones, where the production function is given by either the AK function or the Cobb–Douglas one, while the utility function can be in a linear or power form. Some open questions and conjectures about the regularity of global solutions of finite horizon optimal economic growth problems are formulated in this paper.

H∞ Optimal control of jump systems over multiple lossy communication channels

Abstract In this paper, the $H_{\infty }$ optimal control problem for a Markovian jump linear system (MJLS) over multiple lossy communication channels is considered. It is assumed that the controller communicates with each actuator through a different communication channel. We solve the $H_{\infty }$ optimization problem for the system with a Transmission Control Protocol (TCP) using the theory of dynamic games and obtain a state feedback controller. The infinite horizon $H_{\infty }$ optimization problem is analysed as a limiting case of the finite horizon optimization problem. Then, we obtain the corresponding state feedback controller and show that it stabilizes the closed-loop system in the face of random packet dropouts. A novel and relaxed condition for stability of the closed-loop system with the saddle-point policy is also presented. Further, we show that a weaker notion of observability is sufficient to establish our results.

On a switching control problem with càdlàg costs

This work addresses a switching control problem under which the cost associated with the changes of regimes is allowed to have discontinuities in time. Our main contribution is to show several characterizations of the optimal cost function as well as the existence of ε-optimal control policies. As a by-product, we also study the existence and uniqueness of solutions of a system of backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact of càdlàg type) and depend itself on the unknown solution. In the last part of the paper, we study the case when an underlying diffusion is part of the dynamic of the system. In this special case, the optimal payoff becomes a weak solution of the HJB system of PDEs with discontinuous obstacles which is of quasi-variational type. This paper is somehow a continuation of the papers [B. Djehiche, S. Hamadène, and A. Popier, A finite horizon optimal multiple switching problem, SIAM J. Control Optim. 48 (2009), pp. 2751–2770; S. Hamadène and M.A. Morlais, Viscosity solutions of systems of PDEs with interconnected Obstacle and switching problem, Appl. Math. Optim. 67 (2013), pp. 163–196.] that consider continuous costs.

Optimal Economic Growth Models with Nonlinear Utility Functions

We study a class of finite horizon optimal economic growth problems with nonlinear utility functions and linear production functions. By using a maximum principle in the optimal control theory and employing the special structure of the problems, we are able to explicitly describe the unique solution via input parameters. Economic interpretations of the obtained results and an open problem about the case where the total factor productivity falls into a bounded open interval defined by the growth rate of labor force, the real interest rate, and the exponent of the utility function are also expressed.

Optimal economic growth problems with high values of total factor productivity

ABSTRACT This paper solves a question raised in the paper of Huong, Yao, and Yen [Optimal processes in a parametric optimal economic growth model. Taiwanese J Math. 2020. Available from: https://doi.org/10.11650/tjm/200203] about optimal economic growth problems with production functions and utility functions being all in the form of AK functions. By using a solution existence theorem from the paper of Huong [Solution existence theorems for finite horizon optimal economic growth problems. Preprint arXiv:2001.03298v2. Submitted] and a maximum principle from the book of Vinter [Optimal control. Boston (MA): Birkhäuser; 2000], we prove that the problem in question has a unique solution and give a comprehensive synthesis of the optimal processes. Our results show that if the value of total factor productivity is enough high and the planning time is short, then expanding the production facility does not lead to a higher total consumption satisfaction of the society. Meanwhile, if the value of total factor productivity is enough high and the planning time is relatively long, then the highest total consumption satisfaction of the society is attained only if the largest expansion of the production facility is made until a special time.

Optimal consumption and portfolio decision with stochastic covariance in incomplete markets

Abstract This paper investigates finite and infinite horizon optimal consumption and portfolio choice problems in incomplete markets with stochastic covariance among stock returns. The stock price dynamics are assumed to follow a multivariate stochastic volatility process. The Hamilton-Jacobi-Bellman (HJB) equation is derived by using dynamic programming method and the closed-form expressions for portfolio–consumption strategies are derived in the case of logarithmic utility. In the case of infinite time horizon, we prove the existence of a classical solution and provide a verification theorem. The closed form approximate solutions to the optimal consumption and portfolio polices for a power utility investor are obtained by solving the same utility maximization problem in two fictitious complete markets. Moreover, we verify the existence and optimality of the solution of the original incomplete market and also provide the upper bound of the utility loss of approximate rules. Finally, a numerical application to an incomplete market with stochastic factors show that the utility loss of the approximation rules is small and the performance of the approximate rules is close to optimal.

Event-triggered receding horizon control via actor-critic design

In this paper, we propose a novel event-triggered near-optimal control for nonlinear continuous-time systems. The receding horizon principle is utilized to improve the system robustness and obtain better dynamic control performance. In the proposed structure, we first decompose the infinite horizon optimal control into a series of finite horizon optimal problems. Then a learning strategy is adopted, in which an actor network is employed to approximate the cost function and an critic network is used to learn the optimal control law in each finite horizon. Furthermore, in order to reduce the computational cost and transmission cost, an event-triggered strategy is applied. We design an adaptive trigger condition, so that the signal transmissions and controller updates are conducted in an aperiodic way. Detailed stability analysis shows that the nonlinear system with the developed event-triggered optimal control policy is asymptotically stable. Simulation results on a single-link robot arm with different noise types have demonstrated the effectiveness of the proposed method.

Optimal Processes in a Parametric Optimal Economic Growth Model

A parametric finite horizon optimal economic growth problem is solved by using the maximum principle for optimal control problems with state constraints in the book by R. Vinter [Optimal Control, Birkhäuser, Boston, 2000; Theorem 9.3.1]. From the obtained results it follows that if the total factor productivity is relatively small, then the expansion of the production facility does not lead to a higher total consumption satisfaction of the society.

Robust min–max optimal control design for systems with uncertain models: A neural dynamic programming approach

The design of an artificial neural network (ANN) based sub-optimal controller to solve the finite-horizon optimization problem for a class of systems with uncertainties is the main outcome of this study. The optimization problem considers a convex performance index in the Bolza form. The dynamic uncertain restriction is considered as a linear system affected by modeling uncertainties, as well as by external bounded perturbations. The proposed controller implements a min–max approach based on the dynamic neural programming approximate solution. An ANN approximates the Value function to get the estimate of the Hamilton–Jacobi–Bellman (HJB) equation solution. The explicit adaptive law for the weights in the ANN is obtained from the approximation of the HJB solution. The stability analysis based on the Lyapunov theory yields to confirm that the approximate Value function serves as a Lyapunov function candidate and to conclude the practical stability of the equilibrium point. A simulation example illustrates the characteristics of the sub-optimal controller. The comparison of the performance indexes obtained with the application of different controllers evaluates the effect of perturbations and the sub-optimal solution.

Multiobjective Dynamic Optimization of Cooperative Difference Games in Infinite Horizon

This article derives necessary and sufficient conditions for Pareto optimal solutions in infinite horizon cooperative difference games of autonomous systems with exponentially discounted performances. First, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> constrained optimal control problems are converted into an unconstrained characterization with mixed endpoint constraints by introducing appropriate auxiliary states. Second, the infinite horizon cooperative difference games are transformed equivalently into an augmented and truncated finite horizon optimization problem by defining two real-valued functions and the necessary conditions are derived based on the maximum principle (MP) of discrete-time type. Moreover, we present sufficient conditions for general nonautonomous systems. Finally, the results developed are employed to address the linear quadratic (LQ) cooperative difference games for fixed as well as arbitrary initial states.

On the Finite Horizon Optimal Switching Problem with Random Lag

We consider an optimal switching problem with random lag and possibility of component failure. The random lag is modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the switching mode. The possibility of failures is modeled by having absorbing components. We show existence of an optimal control for the problem by applying a probabilistic technique based on the concept of Snell envelopes.

A finite horizon optimal switching problem with memory and application to controlled SDDEs

We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.

Peak Load Reduction in a Smart Building Integrating Microgrid and V2B-Based Demand Response Scheme

In this paper, we study the peak load reduction in a smart building integrating microgrid and present a comprehensive finite-horizon optimization problem formulated as a dual tracking control problem subject to the quadratic cost function. The smart building integrates a microgrid consisting of photovoltaic modules, battery bank, and plug-in electric vehicles (PEVs). The objective of the microgrid is to satisfy the PEVs load as well as to reduce the peak load of the building. The control strategy aims to take advantages of the microgrid to pay the minimum billing demand to the main grid according to the subscribed power rating. These objectives can be reached through exploiting power generated onsite, local energy storage system, and the operational flexibilities of the PEVs via the vehicle-to-building alternative. In this paper, we focus on the fundamental issue of exploring smart charge and discharge scheduling of PEVs to reduce the peak load of the building. The proposed model has been extensively simulated and tested through case studies using measured solar and load data under the Quebec electricity pricing.

A necessary and sufficient RHC stabilizability condition for stochastic control with delayed input

We study the mean-square stabilizability of Itô stochastic systems with delayed input. In contrast to most previous results, where only sufficient condition was derived, this paper proposes a necessary and sufficient stabilization condition by exploring the receding horizon control (RHC) and the coupled Riccati equations. It is shown that the system controlled by RHC is stabilizable if and only if the two coupled Lyapunov type inequalities are satisfied. The key is to design a novel cost function based on the optimal cost of finite horizon optimization problem.

Model Predictive Control Based Ramp Minimization in Active Distribution Network Using Energy Storage Systems

The growing integration of renewable energy sources, especially the residential photovoltaic (PV) systems, in the distribution networks (DNs) aggravates the ramp-events in transmission system. To address this issue, we propose a novel look ahead dispatch model for the ramp minimization in DNs using distributed energy storage systems (ESSs). The dispatch problem considering the scheduling of ESSs is modeled as a finite-horizon optimization problem and is carried out using model predictive control (MPC) method that takes both current and future information into account. In addition, the optimal power flow in DN is formulated as a second-order cone programing problem to guarantee the global optimality. Numerical results on IEEE 37-bus distribution network show that our proposed model not only brings about 74% reduction of maximum ramp but also yields the near-minimum system operating cost.

Finite-horizon optimal consumption and investment problem with a preference change

In this paper we propose a duality approach to solving an optimal consumption-portfolio selection problem in which an economic agent can choose (irreversible) the preference change time in finite horizon. We assume that after preference change, the agent's coefficient of relative risk aversion becomes higher than before. We use the martingale method and study the variational inequality or free boundary problem arising from the dual problem. Based on partial differential equation (PDE) theory, we analytically characterize the variational inequality and establish the duality relationship. Specifically, we prove that the agent will change his/her preference as soon as the agent's wealth reaches the threshold of wealth as a function of time.