Sufficient Conditions For Optimal Control Research Articles

Necessary and sufficient conditions for optimal control of semilinear stochastic partial differential equations

Necessary and sufficient conditions for optimal control of semilinear stochastic partial differential equations

Optimal control for the complication of Type 2 diabetes: the role of awareness programs by media and treatment.

T2 diabetes is a silent killer and serious public health issue across the world, though awareness of diabetes allows understanding of the causes and prevention of the disease. With this inspiration, we formulate a deterministic model by incorporating awareness and saturated treatment function of the T2 diabetes model to study the dynamics of the disease. We have carried out thoroughly analysis of the model system, including positivity of solutions, boundedness, equilibrium, and stability analysis. Again, we consider the deterministic model system as an optimal control problem by taking awareness (M) and treatment (u) as time-depended control parameters. The sufficient conditions for optimal control for T2 diabetes are obtained utilizing the Pontryagin’s maximum principle in time-dependent controls to find optimal strategies for disease control. We intended to assess the efficacy and costs of several strategies to determine which is the best cost-effective strategy with the limited resources for treatment. The parameters incident rate (beta ), awareness coefficient (p), media (M), and treatment (u) highly influence the dynamics of T2 diabetes. Numerical simulations suggest that both awareness and treatment controls have a significant impact on the optimal system and are economically feasible to reduce the prevalence of T2 diabetes.

Optimal control strategy for cancer remission using combinatorial therapy: A mathematical model-based approach

Abstract In this article, we develop and analyze a non-linear mathematical model of tumor-immune interactions with combined therapeutic drug and treatment controls. To understand under what circumstances the cancerous cells can be destroyed, an optimal control problem is formulated with treatments as control parameters. By designing a quadratic control based functional, we establish the optimal treatment strategies that maximize the number of immune-effector cells, minimize the number of cancer cells, and detrimental effects caused by the amount of drugs. The necessary and sufficient conditions for optimal control are established. We prove the existence and uniqueness of an optimal control problem. To recognize significant system’s parameters, sensitivity analysis are performed for the drug administration and cost functional respectively. We also carry out a cost-effectiveness analysis to determine the most cost-effective therapeutic strategy. The numerical results validate analytical findings and also elucidates that the combinatorial drug therapy can alleviate the cancerous cells under different scenarios.

Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions

In this paper, we investigate a class of impulsive Katugampola fractional differential equations with nonlocal conditions in a Banach space. First, by using a fixed-point theorem, we obtain the existence results for a class of impulsive Katugampola fractional differential equations. Secondly, we derive the sufficient conditions for optimal controls by building approximating minimizing sequences of functions twice.

Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations.

In this paper, we consider the problem of controlling a diffusion process pertaining to an opioid epidemic dynamical model with random perturbation so as to prevent it from leaving a given bounded open domain. In particular, we assume that the random perturbation enters only through the dynamics of the susceptible group in the compartmental model of the opioid epidemic dynamics and, as a result of this, the corresponding diffusion is degenerate, for which we further assume that the associated diffusion operator is hypoelliptic, i.e., such a hypoellipticity assumption also implies that the corresponding diffusion process has a transition probability density function with strong Feller property. Here, we minimize the asymptotic exit rate of such a controlled-diffusion process from the given bounded open domain and we derive the Hamilton-Jacobi-Bellman equation for the corresponding optimal control problem, which is closely related to a nonlinear eigenvalue problem. Finally, we also prove a verification theorem that provides a sufficient condition for optimal control.

Synthesis of Optimal Control-Systems with a Change of the Models of Motion

In this paper, the problem of synthesizing an optimal system controlled with switches is considered. The term switching is understood more broadly than in hybrid systems; i.e., there is a controlled change of the mathematical model of the controlled system at a switching moment: the equations of motion, the state space, and controls are changed. The quality of the control process is estimated by the functional that takes into account both the losses in the continuous motion of the system between switching moments and the cost of each switch. The number of switches and the switching moments themselves are not preassigned and are found during the process of optimization. The sufficient conditions for optimal control are obtained and their application is demonstrated by academic examples of group performance.

Optimal controls for a class of impulsive fractional differential equations with nonlocal conditions

In this paper, we investigate a class of impulsive fractional differential equations with nonlocal conditions in a Banach space. Firstly, we utilize a fixed point theorem to obtain the existence of solution. Secondly, we derive the sufficient conditions for optimal controls by building approximating minimizing sequences of functions twice.

Infinite horizon optimal control problem of mean-field backward stochastic delay differential equation under partial information

This paper investigates an optimal control of an infinite horizon system governed by mean-field backward stochastic differential equation with delay and partial information. Firstly, we establish the existence and uniqueness results for a mean-field backward stochastic differential equation (BSDE) with average delay. Then a class of mean-field time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint equations via duality relation. Meanwhile, necessary and sufficient conditions for optimal control under partial information on infinite horizon are derived. Finally, we apply the theoretical results to study linear-quadratic control problem on infinite horizon to obtain the optimal control, which is explicitly expressed by the solution of a mean-field forward-backward stochastic differential filtering equation.

Stochastic maximum principle for optimal control problems of forward-backward delay systems involving impulse controls

This paper is concerned with the optimal control problems of forward-backward delay systems involving impulse controls. The authors establish a stochastic maximum principle for this kind of systems. The most distinguishing features of the proposed problem are that the control variables consist of regular and impulsive controls, both with time delay, and that the domain of regular control is not necessarily convex. The authors obtain the necessary and sufficient conditions for optimal controls, which have potential applications in mathematical finance.

Maximum Principle for Optimal Control Problems of Forward-Backward Regime-Switching Systems Involving Impulse Controls

This paper is concerned with optimal control problems of forward-backward Markovian regime-switching systems involving impulse controls. Here the Markov chains are continuous-time and finite-state. We derive the stochastic maximum principle for this kind of systems. Besides the Markov chains, the most distinguishing features of our problem are that the control variables consist of regular and impulsive controls, and that the domain of regular control is not necessarily convex. We obtain the necessary and sufficient conditions for optimal controls. Thereafter, we apply the theoretical results to a financial problem and get the optimal consumption strategies.

An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator

In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

Sufficient conditions of optimal control based on Pontryagin’s minimum principle for use in hybrid electric vehicles

Over 10 years ago, the equivalent consumption minimization strategy was introduced as an effective approach, using the concept of equivalent fuel consumption for electricity use, to solve a control problem for hybrid electric vehicles. Although numerous studies have documented outstanding results as a consequence of applying the concept and have shown that the equivalent consumption minimization strategy could be explained on the basis of an optimal control concept such as Pontryagin’s minimum principle, few studies have proven, mathematically, its optimal performance when solving the control problem of hybrid electric vehicles. The present research builds upon previous research studies that proved that the control based on Pontryagin’s minimum principle can be a global optimal solution for hybrid electric vehicles under the assumption that the battery efficiency is not a function of the state of charge. In this paper, we expand upon the original concept, deriving the optimality within more generalized cases than previously reported. In conclusion, if the battery efficiency is a concave function of the state of charge, which is possibly a natural characteristic of the battery, the optimal control based on Pontryagin’s minimum principle enables optimal performance to be achieved. We can therefore apply this control concept to hybrid electric vehicles which use a wide range of states of charge, such as plug-in hybrid electric vehicles.

Continuous-time optimal control for switched positive systems with application to mitigating viral escape

The optimal control problem for a particular class of switched systems is addressed in this paper. Using a linear co–positive cost function, a necessary and sufficient condition for optimal control is derived. Optimal states and costates can lie on a sliding surface, and this corresponds to a chattering switching law. Due to the complexity of exact solution of the general optimal control problem, we introduce a suboptimal, guaranteed cost algorithm, associated with the optimal problem. These results are then applied to a simplified model of HIV viral mutation dynamics, which under simplifying assumptions can be viewed as a positive switched linear system. Simulations compare the optimal switching control law with the sub-optimal guaranteed cost approach.

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel’s martingales and an independent multi-dimensional Brownian motion. Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables. As an application, the linear quadratic stochastic optimal control problem is studied.

Optimal Controls of 3-Dimensional Navier--Stokes Equations with State Constraints

This work is concerned with the maximum principles for optimal control problems governed by 3-dimensional Navier--Stokes equations. Some types of state constraints (time variables) are considered.

Optimal control for quasi-static problem with elastic boundary conditions

We consider the quasi-static problem of viscoelastic body occupying a bounded region Ω ε R3 and subject to an elastic force on the boundary, with these restrictions are imposed on the form ofthe surface and on the boundary conditions, the solution of the problem obtained with the help of appropriate integral equations. We consider the boundary control of the viscoelastic body, and we flnd the necessary and sufficient conditions of optimal control for quasi-static problem.

Periodic maintenance and repair rate control in stochastic manufacturing systems

In this paper, we consider a periodic preventive maintenance, repair, and production model of a flexible manufacturing system with failure-prone machines, where the control variables are the repair rate and production rate. We use periodic preventive maintenance to reduce the machine failure rates and improve the productivity of the system. One of the distinct features of the model is that the repair rate is adjustable. Our objective is to choose a control process that minimizes the total cost of inventory/shortage, production, repair, and maintenance. Under suitable conditions, we show that the value function is locally Lipschitz and satisfies an Hamilton-Jacobi-Bellman equation. A sufficient condition for optimal control is obtained. Since analytic solutions are rarely available, we design an algorithm to approximate the optimal control problem. To demonstrate the performance of the numerical method, an example is presented.

Minimax production planning in failure-prone manufacturing systems

Considers a minimax production planning model of a flexible manufacturing system with machines that are subject to random breakdown and repair. The objective is to choose the rate of production that minimizes the related minimax cost of production and inventory/shortage. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Isaacs (HJI) equation. Under certain conditions, it is shown that the value function is continuously differentiable. A verification theorem is given to provide a sufficient condition for optimal control. Finally, two examples are solved explicitly.

Robust production and maintenance planning in stochastic manufacturing systems

In this paper the authors consider a preventive maintenance and production model of a flexible manufacturing system with machines that are subject to breakdown and repair. The preventive maintenance can be used to reduce the machine failure rates and improve the productivity of the system. The control variables are the rate of maintenance and the rate of production; the objective is to choose a control process that optimizes a robust cost of inventory/shortage, production, and maintenance. The value function is shown to be locally Lipschitz and to satisfy a Hamilton-Jacobi-Isaacs equation. A sufficient condition for optimal control is obtained. Finally, an algorithm is given for solving the optimal control problem numerically.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Lightweight glass mat reinforcement

In this article, we develop and analyze a non-linear mathematical model of tumor-immune interactions with combined therapeutic drug and treatment controls. To understand under what circumstances the cancerous cells can be destroyed, an optimal control problem is formulated with treatments as control parameters. By designing a quadratic control based functional, we establish the optimal treatment strategies that maximize the number of immune-effector cells, minimize the number of cancer cells, and detrimental effects caused by the amount of drugs. The necessary and sufficient conditions for optimal control are established. We prove the existence and uniqueness of an optimal control problem. To recognize significant system’s parameters, sensitivity analysis are performed for the drug administration and cost functional respectively. We also carry out a cost-effectiveness analysis to determine the most cost-effective therapeutic strategy. The numerical results validate analytical findings and also elucidates that the combinatorial drug therapy can alleviate the cancerous cells under different scenarios.