Optimal Control Pair Research Articles

Optimal control results for second‐order semilinear integro‐differential systems via resolvent operators

AbstractIn the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings.

Hilfer-Katugampola fractional stochastic differential inclusions with Clarke sub-differential

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of stochastic Hilfer-Katugampola fractional differential inclusions (SHKFDIs) with non-instantaneous impulsive (NIIs) that is strengthened by Brownian motion (BM) and the Clarke sub-differential. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using stochastic analysis, the Clarke sub-differential's characteristics, and the multi-valued fixed point theorem. Subsequently, by employing Balder's theorem, the existence of optimal control pairs for the considered system is investigated. Eventually, an example is provided to validate the obtained results.

A class of time-optimal feedback control for fractional neutral evolution hemivariational inequalities with fixed delay

ABSTRACT In this paper, we study a class of optimal feedback control for a neutral fractional system with hemivariational inequalities. Initially, we derive some existence results of mild solutions for the system by applying the Marteli's fixed point theorem of multivalued maps, properties of generalized Clarke's subdifferential and semigroup operators. Then, by using the Filippov theorem and the Cesari property, a new set of sufficient conditions are formulated to ensure the existence results of a feasible pair for the feedback control systems. Also, we apply our main results to obtain the optimal feedback control pair. Further, we investigate the time-optimal feedback control for our control system. In the end, an example is given to illustrate our theory.

Mathematical modelling and optimal control analysis of pandemic dynamics as a hybrid system

The study of epidemics using mathematical modelling is critical in understanding its dynamics and proposing potential control measures. We propose a generalised epidemiological model corresponding to a pandemic wherein its dynamics is represented as a novel hybrid system obtained by coupling a deterministic model with a stochastic model. The hybrid system dynamics is established in individualistic (macroscopic) and intra-individualistic (microscopic) scales. The established hybrid system is then considered the basis for an optimal control problem, with the rate of vaccination and velocity of spatial dynamics taken as the control parameters affecting the system’s trajectory. We define the cost functional constituted by the continuous cost corresponding to the deterministic model and discrete costs corresponding to the transitions in the microscopic scale. The objective of the control problem is to find an optimal control pair of vaccination rate and spatial velocity, which minimises the cost functional. We use the Dynamic Programming Principle (DPP) as the optimisation technique, followed by verification of the value function obtained by DPP as a viscosity solution of the appropriate Hamilton–Jacobi–Bellman equation to analyse the existence of an optimal control pair to the hybrid system. We prove the existence of optimal controls to the multi-scale dynamics for pandemic modelling, along with an abstract method to synthesise it.

New discussion on optimal feedback control for Caputo fractional neutral evolution systems governed by hemivariational inequalities

The main focus of this article is to investigate the existence of feedback optimal control for the neutral fractional evolution systems in Hilbert spaces in the sense of the Caputo fractional derivatives. In order to establish the necessary conditions for the proposed problem, we apply the semigroup property, the fixed point theorem of multivalued maps, and the properties of generalized Clarke's subdifferentials. Then, by using the Filippov theorem and the Cesari property, a set of sufficient conditions is formulated to ensure the existence of a feasible pair for the feedback control systems. Finally, we apply our main results to obtain the optimal feedback control pair. In the end, an example is given to illustrate our theory.

An analysis on the optimal control results for second-order Sobolev-type delay differential inclusions of Clarke’s subdifferential type

In this paper, we study the existence and optimal control results for second-order Sobolev-type delay systems with Clarke’s subdifferential type. Initially, the existence of mild solution is established for the proposed second-order delay differential system with the novel ideas of Clarke’s subdifferential. The fixed point theorem of condensing multivalued maps, the strongly continuous cosine family, and the properties of Clarke’s subdifferential are used to establish the existence of mild solution. Moreover, the existence of optimal control pair that is governed by the presented system is verified through Balder’s theorem. Finally, an example is provided to illustrate the main results.

Sliding dynamics and optimal control of avian influenza model with saturated incidence rate

AbstractIn this paper, we develop a susceptible‐infective‐susceptible‐infected‐recovered avian influenza model with saturated incidence rate and three control measures, which is proposed to evaluate the comprehensive effects of different control measures on the spread of avian influenza. Based on the threshold policy, the threshold level is used as the reference index of whether to take control measures. No control measures will be taken when the total number of infected birds and humans is less than the threshold level. Otherwise, the three control measures will be implemented simultaneously to mitigate the spread of the disease. Furthermore, by using the Utkin equivalent control method, the existence of sliding mode equation and its dynamics are investigated. Additionally, the method of dynamic programming is introduced to study the optimal control for the model. Through constructing and proving the existence of viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation, the optimal control pair is further obtained. Finally, numerical simulations are performed to illustrate the corresponding theoretical analysis results.

An Analysis on the Existence of Mild Solution and Optimal Control for Semilinear Thermoelastic System

In this article, the main objective is the conversation about the optimal control problem of the semilinear thermoelastic system, in which the control term is placed solely in the thermal equation. We discuss the existence and uniqueness of mild solutions by applying the contraction mapping for the considered system. By assuming some conditions specified Lagrange’s problem acknowledges at least one optimal control pair. For proving the main results, we are assuming the Lipschitz condition on the nonlinear term.

Optimal control and feedback control for nonlinear impulsive evolutionary equations

The aim of this paper is to study the existence, optimal control and feedback control for a class of nonlinear impulsive evolution equations involving weakly continuous operators. We first obtain the existence result of the evolution equation without impulse in any subinterval by Rothe method, and construct a solution of impulsive evolution equation by using this result. Moreover, we obtain the existence of optimal control pairs for the optimal control problem and feasible pairs for the feedback control system by establishing several sufficient assumptions. An application on a quasistatic frictional contact problem is provided.

Fractional neutral stochastic integrodifferential equations with Caputo fractional derivative: Rosenblatt process, Poisson jumps and Optimal control

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic integrodifferential equations driven by Rosenblatt process and Poisson jumps in Hilbert spaces. First we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive approximation approach. The results are formulated and proved by using the fractional calculus, solution operator and stochastic analysis techniques. The existence of optimal control pairs of system governed by fractional neutral stochastic differential equations driven by Rosenblatt process and poisson jumps is also been presented. An example is provided to illustrate the theory.

Optimal Control Results for Sobolev-Type Fractional Stochastic Volterra-Fredholm Integrodifferential Systems of Order ϑ ∈ (1, 2) via Sectorial Operators

The objective of this paper is to investigate the optimal control results for Sobolev-type fractional stochastic Volterra-Fredholm integrodifferential systems of order with sectorial operators in Hilbert spaces. Initially, we prove the existence of mild solutions by using fractional calculus, stochastic analysis theory, and the Schauder’s fixed point theorem. Next, we demonstrate the existence of optimal control pairs for the given system. Finally, an example is included to show the applications of the developed theory.

Online Optimal Event-Triggered H∞ Control for Nonlinear Systems With Constrained State and Input

Taking safety and performance into consideration, the state and control input of the actual engineering system are often constrained. For this kind of problem, this article puts forward an online dual event-triggered (ET) adaptive dynamic programming (ADP) optimal control algorithm for a class of nonlinear systems with constrained state and input. First, the original system is transformed into another system through the barrier function, after that, a suitable value function with a nonquadratic utility function is designed to obtain the optimal control pair. In addition, on the premise of the asymptotic stability of the system, the trigger condition is devised, and the intersampling time analysis is proved that the algorithm can avoid the Zeno phenomenon. What is more, the critic, action, and disturbance neural networks (NNs) are trained to approximate value function and control sequences, subsequently, the approximation error is proved to be uniformly ultimately boundedness (UUB). Finally, two comparative experiments based on the robot arm model are simulated to verify that the algorithm can make control policies update only when the system has the requirement and keep satisfactory control effect, which can effectively decrease the number of data transfers and reduce the calculation burden.

DYNAMICS OF A DENGUE FEVER MODEL WITH UNREPORTED CASES AND ASYMPTOMATIC INFECTED CLASSES IN SINGAPORE, 2020

This study is devoted to consider a novel model of dengue fever with unreported cases and asymptomatic infected classes, where infected humans is admitted to general and intensive hospitals for treatment. First, the basic reproduction number is calculated by using the next generation matrix method. The disease-free equilibrium is locally asymptotically stable when the basic reproduction number is less than one, but forward bifurcation occurs at the disease-free equilibrium when the basic reproduction number equals one. Then, the endemic equilibrium is consistent persistence when the basic reproduction number is more than one. Next, the existence of the optimal control pair is analyzed, and the mathematical expression of the optimal control is given by using Pontriagin's maximum principle. Finally, based on the dengue fever data in Singapore during the 15-52 weeks of 2020, the best fitting parameters of the model are determined by using Markov Chain Monte Carlo algorithm. The basic reproduction number is 1.6015 (95%CI:(1.5425-1.6675)). Numerical simulation and sensitivity analysis of several parameters are carried out. It is suggested that patients with dengue fever should report and receive treatment in time, which is of great significance for prevention and control of dengue fever.

Reinforcement learning for exploratory linear-quadratic two-person zero-sum stochastic differential games

In this paper, we study an entropy-regularized continuous-time linear-quadratic two-person zero-sum stochastic differential game problem from the perspective of reinforcement learning (RL). By the solvability of a discounted algebraic Riccati equation, we construct a Gaussian closed-loop optimal control pair for the problem, which achieves the best tradeoff between exploration and exploitation. Then, in this exploratory framework, we propose an RL algorithm that relies on only partial system information to solve a stochastic H∞ control problem. The corresponding convergence analysis and simulation examples are also provided to verify the efficiency of the proposed algorithm.

Adaptive dynamic event-triggered control for constrained modular reconfigurable robot

Compared with traditional robot, modular reconfigurable robot (MRR) has the advantages of strong environmental adaptability and flexible task completion. According to the optimal tracking control problem (OTCP) of MRR under some restricted conditions, this paper puts forward a constrained dynamic event-triggered control (DETC) for MRR system with disturbance through adaptive dynamic programming (ADP), which can minimize the information interaction quantity under the premise of system stability and expected control effect. In view of the uncertainty of model coupling part, the identification network is used to estimate the dynamics of MRR and the estimation error is proved to be uniformly ultimate bounded (UUB). The other three groups of critic, action and disturbance neural networks (NNs) are established by the approximation principle of ADP. The optimal control pair is obtained through policy iteration (PI) with DETC, and the triggering condition is designed based on the asymptotic stability of MRR system. At last, the strengths of the algorithm in this paper are validated through simulation experiments.

Optimal Immunotherapy of Oncolytic Viruses and Adopted Cell Transfer in Cancer Treatment

We investigate therapeutic effects of monotherapy of oncolytic viruses, of adopted cell transfer, as well as the two combined therapies over a short time treatment period by applying optimal control techniques. The goal is to minimize the number of susceptible tumor cells and the costs associated with the therapy over the treatment period. We verify that there exists an optimal control pair and derive the necessary conditions. The optimality system is solved numerically to provide optimal protocols under different scenarios with respect to initial tumor sizes and parameter values. Although the two types of therapy do not work synergistically when the viral killing rate by immune cells is large, a small anti-viral killing can improve therapy success of either monotherapy of oncolytic viruses or combined therapy of oncolytic viruses and adopted T cell transfer. This finding can be accomplished either by manipulating certain genes of viruses via genetic engineering or by chemical modification of viral coat proteins to avoid detection by the immune cells.

Optimal feedback control for a class of fractional evolution equations with history-dependent operators

In this paper, we will study optimal feedback control problems derived by a class of Riemann-Liouville fractional evolution equations with history-dependent operators in separable reflexive Banach spaces. We firstly introduce suitable hypotheses to prove the existence and uniqueness of mild solutions for this kind of Riemann-Liouville fractional evolution equations with history-dependent operators. Then, by introducing a feedback iterative technique and applying Filippov theorem, we show the existence of feasible pairs and optimal control pairs of the optimal feedback control systems with history-dependent operators. Finally, we give some applications to illustrate our main results.

Optimal control for semilinear integrodifferential evolution equations in Banach spaces

In this paper, the optimal controls for semilinear integrodifferential evolution equations are investigated in Banach spaces. We demonstrate the existence of feasible pairs by utilising the Cesari condition and the Filippov Theorem. Furthermore, a result on the existence of optimal control pairs for the Lagrange issue is presented.

Optimal feedback control for fractional evolution equations with nonlinear perturbation of the time-fractional derivative term

We study the optimal feedback control for fractional evolution equations with a nonlinear perturbation of the time-fractional derivative term involving Caputo fractional derivatives with arbitrary kernels. Firstly, we derive a mild solution in terms of the semigroup operator generated by resolvents and a kernel from the general Caputo fractional operators and establish the existence and uniqueness of mild solutions for the feedback control systems. Then, the existence of feasible pairs by applying Filippov’s theorem is obtained. In addition, the existence of optimal control pairs for the Lagrange problem has been investigated.

Mathematical analysis, forecasting and optimal control of HIV/AIDS spatiotemporal transmission with a reaction diffusion SICA model

<abstract><p>We propose a mathematical spatiotemporal epidemic SICA model with a control strategy. The spatial behavior is modeled by adding a diffusion term with the Laplace operator, which is justified and interpreted both mathematically and physically. By applying semigroup theory on the ordinary differential equations, we prove existence and uniqueness of the global positive spatiotemporal solution for our proposed system and some of its important characteristics. Some illustrative numerical simulations are carried out that motivate us to consider optimal control theory. A suitable optimal control problem is then posed and investigated. Using an effective method based on some properties within the weak topology, we prove existence of an optimal control and develop an appropriate set of necessary optimality conditions to find the optimal control pair that minimizes the density of infected individuals and the cost of the treatment program.</p></abstract>