Control Theory Techniques Research Articles

Minimizing the penalized goal-reaching probability with multiple dependent risks

We consider a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who wishes to minimize the probability that the value of the wealth process reaches a low barrier before a high goal. We assume that the insurer can purchase per-loss reinsurance for every class of insurance business and invest its surplus in a risk-free asset and a risky asset. Using the technique of stochastic control theory and solving the associated Hamilton-Jacobi-Bellman (HJB) equation, we derive the robust optimal investment-reinsurance strategy and the associated value function. We conclude that the robust optimal investment-reinsurance strategy coincides with the one without model ambiguity, but the value function differs. We also illustrate our results by numerical examples.

An information-theoretic framework for conditional causality analysis of brain networks.

Identifying directed network models for multivariate time series is a ubiquitous problem in data science. Granger causality measure (GCM) and conditional GCM (cGCM) are widely used methods for identifying directed connections between time series. Both GCM and cGCM have frequency-domain formulations to characterize the dependence of time series in the spectral domain. However, the original methods were developed using a heuristic approach without rigorous theoretical explanations. To overcome the limitation, the minimum-entropy (ME) estimation approach was introduced in our previous work (Ning & Rathi, 2018) to generalize GCM and cGCM with more rigorous frequency-domain formulations. In this work, this information-theoretic framework is further generalized with three formulations for conditional causality analysis using techniques in control theory, such as state-space representations and spectral factorizations. The three conditional causal measures are developed based on different ME estimation procedures that are motivated by equivalent formulations of the classical minimum mean squared error estimation method. The relationship between the three formulations of conditional causality measures is analyzed theoretically. Their performance is evaluated using simulations and real neuroimaging data to analyze brain networks. The results show that the proposed methods provide more accurate network structures than the original approach.

A Cortical-Inspired Contour Completion Model Based on Contour Orientation and Thickness.

An extended four-dimensional version of the traditional Petitot-Citti-Sarti model on contour completion in the visual cortex is examined. The neural configuration space is considered as the group of similarity transformations, denoted as M=SIM(2). The left-invariant subbundle of the tangent bundle models possible directions for establishing neural communication. The sub-Riemannian distance is proportional to the energy expended in interneuron activation between two excited border neurons. According to the model, the damaged image contours are restored via sub-Riemannian geodesics in the space M of positions, orientations and thicknesses (scales). We study the geodesic problem in M using geometric control theory techniques. We prove the existence of a minimal geodesic between arbitrary specified boundary conditions. We apply the Pontryagin maximum principle and derive the geodesic equations. In the special cases, we find explicit solutions. In the general case, we provide a qualitative analysis. Finally, we support our model with a simulation of the association field.

Time-consistent reinsurance-investment games for multiple mean-variance insurers with mispricing and default risks

This paper studies a non-zero-sum stochastic differential game for multiple mean-variance insurers. Insurers can purchase proportional reinsurance and invest in a risk-free asset, a market index, a defaultable bond and multiple pairs of mispriced stocks. The dynamics of the mispriced stocks satisfy a “cointegrated system” where the expected returns follow the mean reverting processes, and the bond is defaultable with a recovering proportional value at default. In particular, we assume that the investment opportunities in mispriced stocks are only available for a few insurers, which is more realistic and in line with the superiority of information in the competitive market. Each insurer's objective is maximizing a function of her terminal wealth and competitors' relative wealth under the mean-variance criterion. Using techniques in stochastic control theory, we establish the extended Hamilton-Jacobi-Bellman equations and obtain the equilibrium strategies. Note that the derived solutions are analytical and time-consistent, and we verify the competitive advantages gained from investment opportunities in mispriced stocks. We represent our results in terms of the M-matrices, which help us prove the existence and uniqueness of the solutions and further explicitly analyze how the crucial arguments in the model affect the equilibrium strategies. Numerical examples with detailed sensitivity analyses are presented to support our conclusions.

Dynamic event-triggered boundary control for exponential consensus of multi-agent systems of impulsive PDEs with switching topology

This paper is concerned with the exponential consensus issue for multi-agent systems (MASs) of impulsive partial differential equations (PDEs) with switching topology. A new observer-based dynamic event-triggered boundary control (DETBC) protocol is designed to realize the exponential consensus objective. By Lyapunov functional method, impulse control theory and inequality analysis technique, some conditions are addressed to ensure that the considered MASs achieve the exponential consensus in terms of linear matrix inequalities (LMIs). In addition, the theoretical analysis with respect to the exclusion of Zeno behaviors is given for the designed DETBC mechanism. Finally, two examples are established to indicate the validity of the obtained results.

A hybrid reinsurance-investment game with delay and asymmetric information

In this paper, we investigate the optimal reinsurance and investment problem in the framework of the hybrid stochastic differential game, which includes a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The stochastic Stackelberg differential subgame considers the leader–follower relationship between a reinsurer and two insurers. The reinsurer, as the leader, prices reinsurance premium and invests its wealth in a financial market that contains a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. On the other side, the two insurers, as the followers, purchase proportional reinsurance from the reinsurer and invest in the same financial market. Also, the competitive relationship between the two insurers is characterized by the non-zero-sum stochastic differential subgame in which each insurer considers the relative performance. By introducing the delay feature into the hybrid game, we are able to obtain the wealth processes depicted by the stochastic delay differential equations. Considering the effect of asymmetric information, we assume the reinsurer and insurers have different levels of information about the financial market. The objective of the reinsurer is to maximize the mean–variance functional of its terminal surplus with delay, while each insurer is to maximize the mean–variance functional of its terminal surplus with delay relative to that of its competitor. By applying techniques in stochastic control theory, we obtain the extended Hamilton–Jacobi–Bellman equations, and then by using the idea of backward induction, we establish the equilibrium strategies and the corresponding value functions. Finally, we include some numerical examples and sensitivity analysis to illustrate the effects of model parameters on the equilibrium strategies.

Application of direct and inverse kinematics and dynamics in motion planning of manipulator links

For the synthesis of manipulators and robots, an accurate analysis of the movements of the individual links is essential. This thesis deals with motion planning of the effector of a multi-linked manipulator. An important topic in this area is the orientation and position of links and kinematic pairs in space. In particular, attention should be paid to the position of their endpoint as well as other significant points. Trajectory planning allows the manipulator to perform complex tasks, such as picking and placing objects or following a particular path in space. Overall, trajectory planning of a multibody manipulator involves a combination of direct and inverse kinematics calculations, as well as control theory and optimization techniques. It is an important process enabling manipulators to perform complex tasks such as assembly, handling and inspection. In the design of robot kinematic structures, simulation programs are currently used for their kinematic and dynamic analysis. The proposed manipulator was first solved by inverse kinematics problem in Matlab. Subsequently, the trajectories of the end-effector were determined in Matlab by a direct kinematics problem. In Simulink, using the SimMechanics library, the inverse problem of dynamics was used to determine the trajectories of the moments.

Games in a foreign exchange market and solutions

Abstract Exchange rate and its related risk management are too important for main participants in foreign exchange markets. There are many approaches developed in the literature for studying risk management say arbitrage detection, say finding replication portfolio. However, in the current paper, arbitrage opportunities are studied using the game theory perspective. This paper proposes different types of games played in a specified foreign exchange market in the presence of three exchange rates. Proposed games are exchange rate games in two cases of no arbitrage and existence of arbitrage, optimal stopping game, the arbitrage game, threshold strategies used in global game and Non-cooperative exchange rate game. Most of cases, the bang-bang rule of optimal control is used for finding the Nash equilibriums (NE). However, simulated and stochastic approximation (SA) solutions are also given. Most highlights of the current paper are: (I) considering two types of arbitrage opportunities, simultaneously, (II) translating arbitrage detection as game theory concepts, (III) solving the problem using techniques of optimal control theory. Finally concluding remarks are proposed.

Are physiological oscillations physiological?

Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.

Global Convergence of Policy Gradient Primal–Dual Methods for Risk-Constrained LQRs

While the techniques in optimal control theory are often model-based, the policy optimization (PO) approach directly optimizes the performance metric of interest. Even though it has been an essential approach for reinforcement learning problems, there is little theoretical understanding of its performance. In this paper, we focus on the risk-constrained linear quadratic regulator (RC-LQR) problem via the PO approach, which requires addressing a challenging non-convex constrained optimization problem. To solve it, we first build on our earlier result that an optimal policy has a time-invariant affine structure to show that the associated Lagrangian function is coercive, locally gradient dominated, and has local Lipschitz continuous gradient, based on which we establish strong duality. Then, we design policy gradient primal-dual methods with global convergence guarantees in both model-based and sample-based settings. Finally, we use samples of system trajectories in simulations to validate our methods.

Adaptive behavior in sonar prey tracking tasks: Empirical data and modeling

Echolocating bats can discriminate small differences in target range (Simmons, 1973) and show a clutter interference zone along the range axis of 8–9 cm (Simmons et al., 1988). Because bats use an active sensing system, they can employ various strategies to mitigate the effects of clutter. We studied clutter rejection behaviors of the big brown bat, Eptesicus fuscus, as it tracked a moving target under controlled conditions. We also created data-driven models using control theory and system identification techniques to evaluate oscillating target motion tracking strategies. We trained bats to rest on a stationary platform and track a moving tethered mealworm in two conditions (1) approaching and passing by stationary clutter objects at varying distances and angular offsets, and (2) moving in sinusoidal patterns in an open room. Data from the first experimental condition revealed adaptive echolocation call features and head movements that enabled target tracking in the presence of clutter. Under the oscillating target condition, we analyzed adaptive echolocation behavior to create a data-driven model that maps changes in target motion with sonar call rate. The results of this study demonstrate the bat’s use of active motor control that separates targets from clutter and to track complex motion trajectories.

A STUDY ON NON-CLASSICAL OPTIMAL CONTROL AND DYNAMIC REGIONAL CONTROLLABILITY BY SCALABILITY OF TUMOR EVOLUTION

In this paper, we are interested in Cauchy problem for a parabolic operator. The main system comes from the modeling of a tumor and describes its evolution over time. From the classical optimal control, we show some results and an optimality system for the considered control problem. It is noteworthy that the classical techniques of the optimal control theory are ineffective or are very much technical for certain systems of parabolic evolution. Thus, we address regional controllability by scalability to get around certain difficulties related to estimations. Starting from the solution of the obtained optimality system, we succeeded in establishing the controllability on dynamic subdomains by area-leveling to solve the optimal control problem for this kind of equations.

Robust optimal proportional reinsurance and investment strategy for an insurer and a reinsurer with delay and jumps

In this paper, we consider a robust optimal proportional reinsurance and investment problem in a model with delay and jumps for an insurer and a reinsurer, who are concerned about model uncertainty in model parameters and aim to develop the robust optimal reinsurance and investment strategy. The surplus process of the insurer is described by a diffusion model (which is an approximation of the classical Crmér-Lundberg model) and the insurer can purchase proportional reinsurance. Assume that the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. Moreover, the insurer and the reinsurer can invest in different risky assets whose price processes are governed by the jump-diffusion models. Specially, the insurer's and the reinsurer's wealth processes are modeled by stochastic differential delay equations by introducing the performance-related capital inflow (or outflow). Taking both the insurer and the reinsurer into account, this paper's target is to maximize the minimal expected product of the insurer's and the reinsurer's exponential utilities of the combination of terminal wealth and average performance wealth. By applying the techniques of stochastic optimal control theory, we build the robust Hamilton-Jacobi-Bellman equation. Furthermore, we obtain the robust optimal strategy and the corresponding value function, which extends those for the existing models to slightly more general versions. Finally, we present some numerical examples to demonstrate the impacts of some model parameters on the optimal strategy.

Application and technique of inventory control theory in pharmaceutical sciences

Application and technique of inventory control theory in pharmaceutical sciences

Aroma synthesis and energy consumption in wine fermentation: a multiobjective optimization approach

Consumption of energy during wine fermentation process is directly linked to the temperature profile, which has also proven to have a major impact on the aromatic composition of the end product. This paper studies the impact of the temperature profile in wine fermentation, and the trade-off between the synthesis of aroma compounds and the energy required to regulate the temperature during the process. To this end, we consider a mathematical model representing the main chemical reactions of the wine fermentation including the synthesis of aromas, and a thermal model able to compute the energy required to follow the temperature profile in the fermenter. The objective is to maximize the aroma concentration in the final product while minimizing the energy required to refrigerate the fermenter. The compromise between the two optimization objectives forms Pareto-optimal front solutions, and different solutions are shown in order to better understand the impact of temperature on the process. The approach intends to highlight the potential of control theory techniques and optimization to tackle the inherent cost/quality trade-offs in wine fermentation process, towards a more sustainable energy-efficient winemaking industry.

Satellite Dynamics Toolbox Library: a tool to model multi-body space systems for robust control synthesis and analysis

The level of maturity reached by robust control theory techniques nowadays contributes to a considerable minimization of the development time of an end-to-end control design of a spacecraft system. The advantage offered by this framework is twofold: all system uncertainties can be included from the very beginning of the design process; the validation and Verification (V&V) process is improved by fast detection of worst-case configurations that could escape to a classical sample-based Monte Carlo simulation campaign. Before proceeding to the control synthesis and analysis, a proper uncertain plant model has to be available in order to push these techniques to their limits of performance. In this spirit, the Satellite Dynamics Toolbox Library (SDTlib) offers many features to model a spacecraft system in a multi-body fashion on SIMULINK. Parametric models can be easily built in a Linear Fractional Transformation (LFT) form by including uncertainties and varying parameters with minimal number of repetitions. Uncertain Linear Time Invariant (LTI) and uncertain Linear Parameter-Varying (LPV) controllers can then be synthesized and analyzed in a straightforward way. The authors present in this article a tutorial, that can be downloaded at https://nextcloud.isae. fr/index.php/s/XDfRfHntejHTmmp, to show how to deal with an end-to-end robust design of a spacecraft mission and to provide to researchers a benchmark to test their own algorithms.

Data-Driven Formation Control for Unknown MIMO Nonlinear Discrete-Time Multi-Agent Systems With Sensor Fault.

A data-driven distributed formation control algorithm is proposed for an unknown heterogeneous non-affine nonlinear discrete-time MIMO multi-agent system (MAS) with sensor fault. For the considered unknown MAS, the dynamic linearization technique in model-free adaptive control (MFAC) theory is used to transform the unknown MAS into an equivalent virtual dynamic linearization data model. Then using the virtual data model, the structure of the distributed model-free adaptive controller is constructed. For the incorrect signal measurements due to the sensor fault, the radial basis function neural network (RBFNN) is first trained for the MAS under the fault-free case, then using the outputs of the well-trained RBFNN and the actual outputs of MAS under sensor fault case, the estimation laws of the unknown fault and system parameters in the virtual data model are designed with only the measured input-output (I/O) data information. Finally, the boundedness of the formation error is analyzed by the contraction mapping method and mathematical induction method. The effectiveness of the proposed algorithm is illustrated by simulation examples.

Optimal reinsurance-investment problem for two competitive or cooperative insurers under two investment patterns

The optimal reinsurance-investment problem for two competitive or cooperative insurers is investigated. Each insurer transfers part of the claims risk via reinsurance and invests the surplus in a financial market consisting of one risk-free asset and two risky assets for increasing wealth. Both the aggregated claims of the two surplus processes and the aggregated jumps of the two risky assets are governed by a common Poisson process. We consider two relationships between the two insurers, i.e., competition and cooperation. In addition, we consider two kinds of investment patterns, called the concentration investment and the diversification investment. The objective of the each insurer is to choose the optimal reinsurance-investment strategy so as to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. By the technique of stochastic control theory, the explicit optimal reinsurance-investment strategies and the optimal value functions are obtained in the two investment patterns for the competition case and the cooperation case, and some special cases. Finally, some numerical experiments are carried out to illustrate the effects of model parameters on the optimal strategies and the optimal value functions, which reveal some interesting phenomena and provide useful guidances for reinsurance and investment in reality.

Memory-Limited Partially Observable Stochastic Control and Its Mean-Field Control Approach.

Control problems with incomplete information and memory limitation appear in many practical situations. Although partially observable stochastic control (POSC) is a conventional theoretical framework that considers the optimal control problem with incomplete information, it cannot consider memory limitation. Furthermore, POSC cannot be solved in practice except in special cases. In order to address these issues, we propose an alternative theoretical framework, memory-limited POSC (ML-POSC). ML-POSC directly considers memory limitation as well as incomplete information, and it can be solved in practice by employing the technique of mean-field control theory. ML-POSC can generalize the linear-quadratic-Gaussian (LQG) problem to include memory limitation. Because estimation and control are not clearly separated in the LQG problem with memory limitation, the Riccati equation is modified to the partially observable Riccati equation, which improves estimation as well as control. Furthermore, we demonstrate the effectiveness of ML-POSC for a non-LQG problem by comparing it with the local LQG approximation.

Robust optimal reinsurance in minimizing the penalized expected time to reach a goal

This paper studies a robust optimal reinsurance problem for an ambiguity-averse insurer, who does not have perfect information in the drift term of insurance process. The objective is to minimize the robust value involving the expected time to reach a given capital level before ruin and a penalization of model ambiguity. By using the techniques of stochastic control theory and exponential transformation, we derive the closed-form expressions of the optimal reinsurance strategy and the associated value function for the risk model with cheap reinsurance. For the non-cheap reinsurance, we prove that there exists a “safe level” such that the optimization problem becomes a trivial one when the initial surplus is below this safe level. Therefore, for this case, we focus on solving the corresponding boundary-value problems when the initial surplus is greater than the safe level, and the value function is obtained explicitly as well. Furthermore, we investigate the influence of model ambiguity in theory. Some properties and numerical examples are also presented to show the impact of model parameters on the optimal results.