Singular Control Research Articles

Variational Inequalities on Unbounded Domains for Zero-Sum Singular Controller vs. Stopper Games

We study a class of zero-sum games between a singular controller and a stopper over a finite-time horizon. The underlying process is a multidimensional (locally nondegenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution in a suitable Sobolev class of a variational inequality of min-max type with an obstacle constraint and a gradient constraint. Although the variational inequality and the game are solved on an unbounded domain, we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game. Funding: A. Bovo was partially supported by the Doctoral Studentship from the University of Leeds.

The Maximality Principle in Singular Control with Absorption and Its Applications to the Dividend Problem

The Maximality Principle in Singular Control with Absorption and Its Applications to the Dividend Problem

The Non-Singular Terminal Sliding Mode Control of Underactuated Unmanned Surface Vessels Using Biologically Inspired Neural Network

Underactuated Unmanned Surface Vessels (USVs) are widely used in civil and military fields due to their small size and high flexibility, and trajectory tracking control is a critical research area for underactuated USVs. This paper proposes a trajectory tracking control strategy using the Biologically Inspired Neural Network (BINN) for USVs to improve tracking speed and accuracy. A virtual control law is designed to obtain the required virtual velocity for trajectory tracking control, in which the velocity error is calibrated to ensure that the position error converges to zero. To observe and compensate for unknown and complex environmental disturbances such as wind, waves, and currents, a nonlinear extended state observer (NESO) is designed. Then, a controller based on Non-singular Terminal Sliding Mode (NTSM) is designed to resolve the problems of singular value and controller chattering and to improve the controller response speed. A BINN is introduced to simplify the process of differentiation, reduce the input values of the initial state, and solve the problem of thruster input saturation. Finally, the Lyapunov stability theory is utilized to analyze the stability of the proposed algorithm. The simulation results show that the proposed algorithm has a higher trajectory tracking accuracy and speed than traditional methods.

Reliability and Feasibility of Linear Mixed Models in Fully Crossed Experimental Designs

The use of linear mixed models (LMMs) is increasing in psychology and neuroscience research In this article, we focus on the implementation of LMMs in fully crossed experimental designs. A key aspect of LMMs is choosing a random-effects structure according to the experimental needs. To date, opposite suggestions are present in the literature, spanning from keeping all random effects (maximal models), which produces several singularity and convergence issues, to removing random effects until the best fit is found, with the risk of inflating Type I error (reduced models). However, defining the random structure to fit a nonsingular and convergent model is not straightforward. Moreover, the lack of a standard approach may lead the researcher to make decisions that potentially inflate Type I errors. After reviewing LMMs, we introduce a step-by-step approach to avoid convergence and singularity issues and control for Type I error inflation during model reduction of fully crossed experimental designs. Specifically, we propose the use of complex random intercepts (CRIs) when maximal models are overparametrized. CRIs are multiple random intercepts that represent the residual variance of categorical fixed effects within a given grouping factor. We validated CRIs and the proposed procedure by extensive simulations and a real-case application. We demonstrate that CRIs can produce reliable results and require less computational resources. Moreover, we outline a few criteria and recommendations on how and when scholars should reduce overparametrized models. Overall, the proposed procedure provides clear solutions to avoid overinflated results using LMMs in psychology and neuroscience.

The second-order maximum principle for partially observed optimal controls

In this paper, the stochastic control system under partial observation for singular optimal controls in the classical sense is studied. Note that both the system and the observation contain noise, in which the control variable is allowed to enter into all the coefficient terms. By adding some regularity assumptions, the pointwise second-order maximum principle under partial observation can be obtained by Malliavin calculus. Eventually, an example is given to demonstrate how to find the optimal control.

Adaptive Fuzzy Finite-Time Singular Perturbation Control for Flexible Joint Manipulators With State Constraints

Adaptive Fuzzy Finite-Time Singular Perturbation Control for Flexible Joint Manipulators With State Constraints

On singular control of reflected diffusions

On singular control of reflected diffusions

Mathematical modelling of COVID-19 transmission with optimal control and cost-effectiveness analysis

The present global health threat is the novel coronavirus disease (COVID-19), caused by a new strain of the SARS-CoV-2 coronavirus. In this study, have employed optimal control theory, aided by Pontryagin’s Maximum Principle, to explore optimal control measures. Specifically, we have investigated time-dependent intervention strategies, including the proper use of personal protective measures and vaccination. Bifurcation analysis was conducted and results shows that the model system exhibit a forward bifurcation. The optimal control system have been numerically simulated using the fourth-order Runge–Kutta methods. The results show that the implementation of the combination of the two interventions was more significant and effective in minimizing the spread of the COVID-19 than the implementation of a single control measure. These findings underscore the significance of multifaceted intervention approaches over singular control measures. Notably, the combined implementation of interventions emerges as markedly more effective in containing COVID-19 transmission. Moreover, our study identifies personal protective measures as a particularly cost-effective intervention, offering substantial relief from the burden of the pandemic within the population. We anticipate that our research will inform evidence-based approaches to pandemic control and aid in the ongoing global efforts to safeguard public health.

Second‐order necessary optimality conditions for discrete‐time stochastic systems

SummaryThis paper deals with the second‐order necessary optimality conditions for discrete‐time stochastic optimal control problems under weakened convexity assumptions. Using a special variation of the control, and by virtue of a new discrete‐time backward stochastic equation, we establish a more general and constructive first‐order necessary optimality condition in the form of a global stochastic maximum principle. Moreover, by introducing a new discrete‐time backward stochastic matrix equation, the second‐order multipoint necessary optimality conditions of singular controls are derived, which covers and improves the classical second‐order necessary optimality conditions of discrete‐time stochastic systems.

A maximum principle for progressive optimal control of mean‐field forward–backward stochastic system involving random jumps and impulse controls

AbstractIn this paper, we study an optimal control problem of a mean‐field forward–backward stochastic system with random jumps in progressive structure, where both regular and singular controls are considered in our formula. In virtue of the variational technology, the related stochastic maximum principle (SMP) has been obtained, and it is essentially different from that in the classical predictable structure. Specifically, there are three parts in our SMP, that is, continuous part, jump part, and impulse part, and they are, respectively, used to characterize the characteristics of the optimal controls at continuous time, jump time, and impulse time. This shows that the progressive structure can more accurately describe the characteristics of the optimal control at the jump time. We also give two linear–quadratic examples to show the significance of our results.

Vectorized Trigonometric Regularization for Singular Control Problems with Multiple State Path Constraints

Vectorized Trigonometric Regularization for Singular Control Problems with Multiple State Path Constraints

Assessment of heavy metal distribution, risk, and sourcing in mangrove sediments from three Saudi Arabian Red Sea locations

This work measured the concentration of heavy metal cations (V, Cr, Ni, Cu, Zn, Cd, and Pb) and associated anionic ligands (SOM, carbonate, and silicate) in mangrove sediment samples from the Red Sea of Saudi Arabia to set a biogeochemical baseline within these as yet poorly constrained sensitive ecosystems. Sediments were collected from three regions along the Saudi Arabian coast: Yanbu, Jeddah, and the Farasan Islands. Risk indices and statistical analyses were applied to assess contamination levels and potential sourcing. Results show that Yanbu is at environmental risk as it shows higher concentrations and anthropogenic signatures for most metals (Cr, Cu, Zn, Cd and Pb) compared to the other two regions. Jeddah metal concentrations are similar to the Islands; however, the statistical analyses suggest that a singular anthropogenic source controls heavy metal delivery to its environment.

Distributed Optimal Consensus Control of Multiagent Systems Involving State and Control Dependent Multiplicative Noise

This paper aims to solve the multi-objective optimization control problem by investigating the consensus problem of multi-agent simultaneous optimization with state and control dependent multiplicative noise. The main contributions of this work are two-fold. First, the optimal consensus control problem is solved by designing the free item of an optimal controller, and the coefficient of the feedback gain is obtained based on analyzing a parameterized generalized algebraic Riccati equation. Second, we derive a new consensus controller that comprises two parts different from the current consensus. One part minimizes a linear quadratic index with a singular control weighting matrix, and the other affords the multi-agent systems to achieve mean square consensus.

Multi-control commercial aircraft trajectory optimization in a vertical plane with state-inequality constraints via singular control theory

We present a computational framework based on singular control theory to optimize commercial aircraft flights in a vertical plane. The optimization problem involves the lift coefficient (a regular control), and throttle setting (a singular control). However, given a small aerodynamic path angle typically associated with commercial aircraft flights and the significant time-scale separation in flight dynamics, it is customary to treat the aerodynamic path angle as a singular control and replace the lift coefficient as a function of states. This results in a model that is singular on both controls. This research sheds light on several key challenges that have either been unexplored, such as the possibility of switches in both controls even during a specified flight phase, or underexplored, such as state-inequality constraints and wind shear, in relation to the pure singular model. Adhering to the Pontryagin’s maximum principle, we develop a switching-point algorithm to solve the pure singular model for the direct operative cost. Extensive analysis is conducted to understand the impact of various relaxed parameters. The simulation results obtained from the pure singular model are validated through first and second order optimality conditions. Finally, we compare the solutions obtained from the pure singular model with those of the original model, ultimately concluding that the pure singular model yields exceptionally accurate results. In conclusion, the paper highlights the computational time and accuracy advantages of the proposed optimization framework. Furthermore, it underscores the noteworthy observation that the sensitivity of the associated Hamiltonian system with respect to aircraft mass is marginal.

Approximation of the infinite-horizon value function of the switched LQR problem

This paper considers the approximation of the infinite-horizon value function of the continuous-time switched linear quadratic regulator (LQR) problem. We propose a two successive approximation approach to achieving arbitrarily high accuracy. The infinite-horizon value function is first approximated by a finite-horizon value function, and the resulting truncation error is shown to converge to zero exponentially as the length of the control horizon increases. Then, the finite-horizon value function is approximated through a uniform discretization of the switched linear system, and the approximation error is proven to be proportional to the discretization step. These theoretical results facilitate the development of an efficient approximation algorithm, which further results in a near-optimal control policy. Compared with the existing methods, our algorithm obviates numerical issues induced by singular optimal control. Two numerical examples are presented to illustrate the effectiveness of our algorithm.

A mixed singular/switching control problem with terminal cost for modulated diffusion processes

In this paper, we study the regularity of the value function associated with a stochastic control problem involving two controls that act simultaneously on a modulated multidimensional diffusion process. The first is a switching control modelling a random clock. Whenever the random clock rings, the generator matrix is replaced by another, leading to a different dynamic for the finite state Markov chain of the modulated diffusion process. The second is a singular stochastic control that is executed on the process within each regime.

Two-Sided Singular Control of an Inventory with Unknown Demand Trend

Two-Sided Singular Control of an Inventory with Unknown Demand Trend

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Stage‐cost design for optimal and model predictive control of linear port‐Hamiltonian systems: Energy efficiency and robustness

AbstractWe consider singular optimal control of port‐Hamiltonian systems with minimal energy supply. We investigate the robustness of different stage‐cost designs w.r.t. time discretization and show that alternative formulations that are equivalent in continuous time, differ strongly in view of discretization. Furthermore, we consider the impact of additional quadratic control regularization and demonstrate that this leads to a considerable increase in energy consumption. Then, we extend our results to the tracking problem within model predictive control and show that the intrinsic but singular choice of the cost functional as the supplied energy leads to a substantial improvement of the closed‐loop performance.

A singular stochastic control problem with direction switching cost

We introduce a new class of singular stochastic control problems in which the process controller not only chooses the push intensity, at a price proportional to the displacement caused by his action, but he can also change the allowable control direction, paying a fixed cost for each such switching. Singular control of the one-dimensional Brownian motion with quadratic instantaneous cost function and costly direction switching on the infinite time horizon is analyzed in detail, leading to a closed-form solution. This example is used as an illustration of qualitative differences between the class of problems considered here and classic singular stochastic control.