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In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the feedback stabilization of an infinite-dimensional semilinear system evolving in reflexive Banach state space. The concept of bounded control is also investigated in the realistic domain. Sufficient conditions for appropriate feedback control to ensure strong and weak stabilization are given. [ABSTRACT FROM AUTHOR]

In this paper, we study the controllability of two problems involving the same Chandrasekhar-type integral equation, but under different kinds of controls. A viability condition is imposed as well. We provide existence results of continuous trajectories coupled to continuous controls. Then, in the non-viable case, we investigate the optimal estimates to be taken in view of the existence of solutions for both problems. The last part of the paper deals with the application of the previous results to the classical Chandrasekhar equation, first showing the existence of a viable continuous solution, then providing also uniqueness and approximability. Two examples of controllability problems governed by this equation are given. [ABSTRACT FROM AUTHOR]

This paper discusses autonomous and nonautonomous epidemic models with nonlinear incidence rate of saturated mass action and feedback controls. The global asymptotic stability of disease-free equilibrium and the endemic equilibrium of the autonomous system is established using suitable Lyapunov functional. It is shown that by choosing suitable values of feedback control variables, one can make the disease endemic or extinct as time evolves. Moreover, the effect of coefficient of inhibition on the persistence of disease is also discussed. We discuss the permanence, existence, uniqueness and asymptotic stability of an almost periodic solution of the model. The analytical results obtained in this paper are illustrated with the help of numerical examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the feedback control problems of a new stochastic ecosystem with regime switching expressed by a partially observable Markov chain. The main idea is to convert the partially observable system into a complete observable one by estimating the unobservable Markov chain by Wonham filter. For the complete observable system, the authors discuss the existence, uniqueness, stochastic boundedness and sample path continuity of the positive solution by constructing Lyapunov functions, and design various forms of feedback controls to regulate the long-time dynamic behaviours of the system. The idea and methods in this paper can also be applied to other partially observable control systems such as financial systems, infections disease system and so on. [ABSTRACT FROM AUTHOR]

This paper proposes a new stochastic eco-epidemiological model with nonlinear incidence rate and feedback controls. First, we prove that the stochastic system has a unique global positive solution. Second, by constructing a series of appropriate stochastic Lyapunov functions, the asymptotic behaviors around the equilibria of deterministic model are obtained, and we demonstrate that the stochastic system exists a stationary Markov process. Third, the conditions for persistence in the mean and extinction of the stochastic system are established. Finally, we carry out some numerical simulations with respect to different stochastic parameters to verify our analytical results. The obtained results indicate that the stochastic perturbations and feedback controls have crucial effects on the survivability of system. [ABSTRACT FROM AUTHOR]

In this paper, we address the question of the construction of a nearly time optimal feedback law for a minimum time optimal control problem, which is robust with respect to internal and external perturbations. For this purpose we take as starting point an optimal synthesis, which is a suitable collection of optimal trajectories. The construction we exhibit depends exclusively on the initial data obtained from the optimal feedback which is assumed to be known. [ABSTRACT FROM AUTHOR]

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model.

In this paper, we propose a discrete pollination mutualism in a plant-pollinator system with the Beddington-DeAngelis functional response and feedback controls. By applying the comparison theorem of a difference equation and constructing some suitable Lyapunov functions, sufficient conditions are obtained for the permanence and the extinction of the system. Moreover, under some suitable conditions, we show that the solution of the system is globally attractive. The paper ends by some numerical simulations and a brief discussion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the output stabilisation for a class of distributed bilinear system evolving in a spatial domain Ω. We give sufficient conditions for strong and weak output stabilisation. Also, the output stabilisation is discussed using a minimisation problem. Examples and simulations are given. [ABSTRACT FROM AUTHOR]

In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback control on predator, the prey species still extinct, whereas the predator species is stable at certain values. Several examples are presented to show the feasibility of the main results. [ABSTRACT FROM AUTHOR]

In this paper, a May cooperative system with feedback controls is proposed and studied. The dynamic behaviors of the system are discussed by using the Lyapunov function method. If $b_{i}\neq0$, $i=1,2$, we show that feedback control variables have no influence on the global stability of the unique positive equilibrium of the system, which means that feedback control variables only change the position of the positive equilibrium and retain its global stability property. If $b_{i}= 0$, $i=1,2$, we can make the system which has a unique globally stable equilibrium or has unboundedly large solutions become globally stable. Some examples are given to illustrate the feasibility of the main results. [ABSTRACT FROM AUTHOR]

This research presents a novel approach to address the complexities of heterogeneous lung cancer dynamics through the development of a Fractional-Order Model. Focusing on the optimization of combination therapy, the model integrates immunotherapy and targeted therapy with the specific aim of minimizing side effects. Notably, our approach incorporates a clever fusion of Proportional-Integral-Derivative (PID) feedback controls alongside the optimization process. Unlike previous studies, our model incorporates essential equations accounting for the interaction between regular and mutated cancer cells, delineates the dynamics between immune cells and mutated cancer cells, enhances immune cell cytotoxic activity, and elucidates the influence of genetic mutations on the spread of cancer cells. This refined model offers a comprehensive understanding of lung cancer progression, providing a valuable tool for the development of personalized and effective treatment strategies. the findings underscore the potential of the optimized treatment strategy in achieving key therapeutic goals, including primary tumor control, metastasis limitation, immune response enhancement, and controlled genetic mutations. The dynamic and adaptive nature of the treatment approach, coupled with economic considerations and memory effects, positions the research at the forefront of advancing precision and personalized cancer therapeutics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete Lotka-Volterra competitive system with the effect of toxic substances and feedback controls. By using the method of discrete Lyapunov function and by developing a new analysis technique, we obtain the sufficient conditions which guarantee that one of the two species will be driven to extinction while the other will be permanent. We improve the corresponding results of Li and Chen [Extinction in two-dimensional discrete Lotka-Volterra competitive system with the effect of toxic substances, Dynam. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 15 (2008) 165-178]. Also, an example together with their numerical simulations shows the feasibility of our main results. It is shown that toxic substances and feedback control variables play an important role in the dynamics of the system. [ABSTRACT FROM AUTHOR]

In this paper, a basic virus dynamics model with feedback controls is considered and analyzed to restrain infected cells. The asymptotic stability of the model is investigated by using the Lyapunov stability theorem and numerical experiments. The obtained results show that the feedback control variables are effective to restrain infected cells in the original virus dynamics model. In particular, we propose two effective strategies for selecting the feedback control variables based on the asymptotic stability of the model with the feedback control variables. A set of numerical experiments is performed to confirm the validity of the theoretical results and the effectiveness of the proposed control strategies. [ABSTRACT FROM AUTHOR]

In this paper, an N-species cooperation system with time delays and feedback controls is proposed on time scales. In this process, some inequalities, which play a vital roles in the proof of main results, are firstly proved. Then, by applying the theory of differential inequality, sufficient conditions which guarantee the permanence of the system are obtained on time scales. Our results improve and complement some known results to some degree in the literature. [ABSTRACT FROM AUTHOR]

In this paper stability and attractivity in non-autonomous time-discrete dynamical systems is investigated with the aid of Lyapunov functions. The results are applied to the problem of stabilization of controlled systems by feedback controls. In the final section of the paper we give sufficient conditions for norm-bounded null-controllability of linear systems. [ABSTRACT FROM AUTHOR]

In this paper, we concern a class of the generalized delayed stochastic predator-prey models with feedback controls based on discrete observations. The existence of global positive solution is given first. Then we discuss the deterministic model briefly, and establish the necessary conditions and the sufficient conditions for almost-sure extinction and persistence in mean for the stochastic system, where we show that the feedback controls can change the properties of the population systems significantly. Finally, numerical simulations are introduced to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem's iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory. [ABSTRACT FROM AUTHOR]

The course of Electric Machinery has been one of the core courses in the Electrical Engineering (EE) curriculum at Texas A&M University—Kingsville (TAMUK). In the past, the course was designed to focus mainly on topics of transformers and electric machines and was typically presented under the assumption of steady-state conditions. Unfortunately, many students considered the course old-fashioned and unexciting, and this lack of interest, in turn, seriously limited their ability to understand and appreciate the wide-ranging applications of electric machines. More important, electric machines today have become widely integrated with power electronics, digital signal processing (DSP), and digital control technology. However, these technological changes were not adequately reflected in the traditional teaching structure of the course. This paper presents a new approach that, by integrating computer-assisted teaching methodology in classroom presentation, substantially restructures the course so that it provides a comprehensive treatment of electric-drive systems with discussions covering electric machines, power electronic converters, feedback controls, power supply systems, and mechanical systems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an SI epidemic model with feedback controls in a patchy environment where individuals in each patch can disperse among n ( n  ≥ 2) patches. We derive the basic reproduction number R 0 and prove that the disease-free equilibrium is globally asymptotically stable if R 0  ≤ 1. In the case of R 0  > 1, we derive sufficient conditions under which the endemic equilibrium is unique and globally asymptotically stable. Our proof of global stability utilizes the method of global Lyapunov functions and results from graph theory. Numerical simulations are carried out to support our theoretical results. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous discrete competition system with nonlinear interinhibition terms and feedback controls. By constructing a suitable Lyapunov function, we obtain some criteria about the extinction of one of the two species and the corresponding feedback controls varieties. Our conclusions not only supplement but also improve some existing ones. Numerical simulations are used to illustrate our analytic analysis. We show that feedback control variables play an important role in the extinction property of the system. [ABSTRACT FROM AUTHOR]

This paper investigates a coupled system of fractional-order differential equations on network with feedback controls (CSFDENFCs). By using the contraction mapping principle, Lyapunov method, graph theoretic approach and inequality techniques, some sufficient conditions are derived to ensure the existence, uniqueness and global Mittag–Leffler stability of the equilibrium point of CSFDENFCs. Finally, numerical simulations are presented to demonstrate the validity and feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

A non-autonomous allelopathic phytoplankton model with nonlinear inter-inhibition terms and feedback controls is studied in this paper. Based on the comparison theorem of differential equation, some sufficient conditions for the permanence of the system are obtained. We study the extinction of one of the species by using some suitable Lyapunov type extinction function. Our analyses extend those of Xie et al. (Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Advances in Difference Equations, 2016, 2016, 258) and show that the feedback controls and toxic substances have no effect on the permanence of the system but play a crucial role on the extinction of the system. Some known results are extended. [ABSTRACT FROM AUTHOR]

A modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response and feedback controls is studied. By applying the differential inequality theory, sufficient conditions which guarantee the permanence of the system are obtained. Our results improve the main results of Zhang et al. (Abstr. Appl. Anal. 2014:252579, 2014). One example is presented to verify our main results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider an -species cooperation system with continuous time delays and feedback controls. By applying a new integral inequality, we obtain a set of sufficient conditions which ensure the permanence of the system. Our result shows that feedback control variables have no influence on the permanence of the system, thus improving the corresponding results of Chen, Liao and Huang [F.D. Chen, X.Y. Liao, Z.K. Huang, The dynamic behavior of -species cooperation system with continuous time delays and feedback controls, Applied Mathematics and Computation 181 (2) (2006) 803–815]. [Copyright &y& Elsevier]

Abstract: This paper discusses a two species discrete model of mutualism with delays and feedback controls. Sufficient conditions are obtained for the permanence of the system. The results show that feedback control variables have no influence on the persistence property of the system. [Copyright &y& Elsevier]

Abstract: In this paper, we study the positive almost periodic solutions for a class of almost periodic Lotka-Volterra type system with delays and feedback controls. Applying Schauder’s fixed point theorem, a criterion on the existence of the positive almost periodic solution of the system is obtained. Our new criterion, which improves and generalizes some well known results, can be easily checked. [Copyright &y& Elsevier]

Abstract: In this paper, a discrete N-species cooperation system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a N-species cooperation system with continuous time delays and feedback controls. For general nonautonomous case, by using differential inequality theory and constructing a suitable Lyapunov functional, we obtain sufficient conditions which guarantee the permanence and the global attractivity of the system. For the periodic case, we obtain sufficient conditions which guarantee the existence, uniqueness and stability of a positive periodic solution. For the almost periodic case, by using the Razumikhin type theorem, we obtain the sufficient conditions which guarantee the existence of a positive almost periodic solution of the system. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a periodic multi-species Kolmogorov type competitive system with delays and feedback controls. Applying Schauder’s fixed point theorem, a criterion on the existence of the positive periodic solution of the system are obtained. As application, the existence of the positive periodic solution of a kind of nonlinear competitive model with delays and feedback controls is discussed. [Copyright &y& Elsevier]

In this paper, the following almost periodic n-species competitive system with feedback controls are studied by using the comparison theorem and constructing suitable Lyapunov function, where some important factors such as the effect of toxic and the age-structure are also considered simultaneously. Some sufficient conditions are obtained for the existence of a unique almost periodic solution of above model. Examples show that the obtained criteria are new, general, and easily verifiable. [Copyright &y& Elsevier]

Abstract: In this paper, an SI epidemic model with feedback controls is first proposed. By constructing a suitable Lyapunov function, global stability of the disease-free equilibrium and the endemic equilibrium of the model is investigated. The results we obtained show that by choosing suitable values of feedback control variables, we can make the disease endemic or extinct. In other words, feedback control variables play an important role in dealing with the disease. An example is presented to verify our main results. [Copyright &y& Elsevier]

In this paper, we prove the necessity of a terminal condition for a solution of the Bellman Equation to be the value function in dynamic optimization problems with unbounded payoffs. We also state the weakest sufficient condition, which can be applied in a large class of problems, including economic growth, resource extraction, or human behaviour during an epidemic. We illustrate the results by examples, including simple linear–quadratic problems and problems of resource extraction, with multiple solutions to the Bellman Equation or the maximizer of the right hand side of the Bellman Equation with the actual value function being the worst control instead of being optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem’s iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory.

The output stabilization for a class of distributed semilinear system is obtained with stabilizing controls that depend on the output operator. Then, we give sufficient conditions for exponential, strong and weak stabilization. The obtained results are illustrated by examples and simulations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback control on predator, the prey species still extinct, whereas the predator species is stable at certain values. Several examples are presented to show the feasibility of the main results.

In a series of previous works, we derived nonlocal necessary optimality conditions for free-endpoint problems. These conditions strengthen the Maximum Principle and are unified under the name “feedback minimum principle”. The present paper is aimed at extending these optimality conditions to terminally constrained problems. We propose a scheme of the proof of this generalization based on a “lift” of the constraints by means of a modified Lagrange function with a quadratic penalty. Implementation of this scheme employs necessary optimality conditions for quasi-optimal processes in approximating optimal control problems. In view of this, in the first part of the work, the feedback minimum principle is extended to quasi-optimal free-endpoint processes (i.e. strengthen the so-called $\varepsilon$-perturbed Maximum Principle). In the second part, this result is used to derive the approximate feedback minimum principle for a smooth terminally constrained problem. In an extended interpretation, the final assertion looks rather natural: If the constraints of the original problem are lifted by a sequence of relaxed approximating problems with the property of global convergence, then the global minimum at a feasible point of the original problem is admitted if and only if, for all $\varepsilon>0$, this point is $\varepsilon$-optimal, for all approximating problems of a sufficiently large index. In respect of optimal control with terminal constraints, the feedback $\varepsilon$-principle serves exactly for realization of the formulated assertion.

A non-autonomous allelopathic phytoplankton model with nonlinear inter-inhibition terms and feedback controls is studied in this paper. Based on the comparison theorem of differential equation, some sufficient conditions for the permanence of the system are obtained. We study the extinction of one of the species by using some suitable Lyapunov type extinction function. Our analyses extend those of Xie et al. (Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Advances in Difference Equations, 2016, 2016, 258) and show that the feedback controls and toxic substances have no effect on the permanence of the system but play a crucial role on the extinction of the system. Some known results are extended.