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We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term. [ABSTRACT FROM AUTHOR]

This paper investigates a discrete conformable fractional order bacteria population model in a microcosm. When the parameters of this discrete model satisfy some conditions, its positive fixed point has a double eigenvalue - 1 with geometric multiplicity 1, corresponding to a 1:2 strong resonance. Applying the normal form theory and the approximation theory by a flow, we firstly transform this discrete model into a two-dimensional ordinary differential system. Then we use the bifurcation theory to analyze the qualitative properties of this ordinary differential system near the degenerate equilibrium. Therefore, the qualitative properties of the corresponding fixed point are obtained, which show that this discrete model undergoes the period-doubling bifurcation, the Neimark-Sacker bifurcation, the homoclinic bifurcation and the double closed orbit bifurcation. We also add a hybrid controller to this model such that the fixed point is locally asymptotically stable. Finally, numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this study, first-order necessary optimality conditions, in the form of a weak maximum principle, are derived for discrete optimal control problems with mixed equality and inequality constraints. Such conditions are achieved by using the Dubovitskii–Milyutin formalism approach. Nondegenerate conditions are obtained under the constant rank of the subspace component (CRSC) constraint qualification, which is an important generalization of both the Mangasarian–Fromovitz and constant rank constraint qualifications. Beyond its theoretical significance, CRSC has practical importance because it is closely related to the formulation of optimization algorithms. In addition, an instance of a discrete optimal control problem is presented in which CRSC holds while other stronger regularity conditions do not. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of the discrete fractional systems of order 1 < α < 2 (∗) C ∇ α u n = A u n + B v n + f (n , u n) , n ≥ 2 , subject to the initial states u 0 = x 0 , u 1 = x 1 , where A is a closed linear operator defined in a Hilbert space X, B is a bounded linear operator from a Hilbert space U into X , f : N 0 × X → X is a given sequence and C ∇ α u n is an approximation of the Caputo fractional derivative ∂ t α of u at t n : = τ n , where τ > 0 is a given step size. To do this, we first study resolvent sequences { S α , β n } n ∈ N 0 generated by closed linear operators to obtain some subordination results. In addition, we discuss the existence of solutions to (∗) and next, we study the existence of optimal controls to obtain the approximate controllability of the discrete fractional system (∗) in terms of the resolvent sequence { S α , β n } n ∈ N 0 for some α , β > 0. Finally, we provide an example to illustrate our results. [ABSTRACT FROM AUTHOR]

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman–Yakubovich–Popov inequalities is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete dynamic control system with noise. It is required that at the terminal moment of process, the phase point is contained in a given ring-shaped set. We find the operator of program absorption, which allows one to formulate conditions for the set of initial positions that guarantee the fulfillment of the required inclusion at a given moment of time. [ABSTRACT FROM AUTHOR]

In the context of constrained control-systems, the Set of Sustainable Thresholds plays in a sense the role of a dual object to the so-called Viability Kernel, because it describes all the thresholds that must be satisfied by the state of the system along a time interval, for a prescribed initial condition. This work aims at analyzing the sensitivity of the Set of Sustainable Thresholds, when it is seen as a set-valued map that depends on the initial position. In this regard, we investigate semicontinuity and Lipschitz continuity properties of this mapping, and we also study several contexts when the Set of Sustainable Thresholds is convex-valued. [ABSTRACT FROM AUTHOR]

This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the ε -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the ε -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the ε -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions. [ABSTRACT FROM AUTHOR]

This article presents an approximation of discrete Markov decision processes with small noise on Borel spaces with an infinite horizon and an expected total discounted cost by the corresponding deterministic Markov process. In both cases, the dynamics evolve through a system consisting of two coupled difference equations. It is assumed that the difference equations of the system are perturbed by a small noise. Under our assumptions, a bound for the stability index is given, and the optimal cost convergence rate is estimated using a small perturbation parameter. Moreover, the convergence of the optimal policy on compact subsets is verified. Finally, two examples are presented to illustrate the developed theory.

We introduce the notions of q-invariance entropy and q-metric invariance entropy for control systems by Carathéodory–Pesin structures. We obtain some basic properties of these entropies and some relations between q-invariance entropies and invariance pressures. Furthermore, we discuss relations between q-metric invariance entropies and local metric invariance entropies. [ABSTRACT FROM AUTHOR]

Three types of BS inner invariance pressures for control systems, as well as three versions of measure-theoretic BS inner invariance pressures, are introduced in this paper. We show that those BS inner invariance pressures are equivalent for the strongly invariant sets. Moreover, it is confirmed that they are the roots of Bowen's equations for control systems. Then, we can further obtain an inverse variational principle for the measure-theoretic BS inner invariance pressure. Finally, a type of upper capacity BS invariance pressure and an explicit formula are presented for computing this invariance pressure of controlled invariant sets under some reasonable assumptions. [ABSTRACT FROM AUTHOR]

We introduce the notions of (inner) equi-invariability, (inner) finite equi-invariability, and bounded (inner) invariance complexity. We show that they are equivalent for (open) compact control sets with nonempty interior. Moreover we show that a control set is either (inner) equi-invariant or (inner) unstable under some technical assumptions. [ABSTRACT FROM AUTHOR]

This work concerns with Markov decision processes on a denumerable state space. It is assumed that the performance of a control policy is measured by the average criterion associated with a risk-seeking controller with constant risk-sensitivity coefficient. The structural assumptions on the model ensure that the optimal average cost is constant, but it is possible that the optimalty equation does not admit a solution. In this context, a risk-sensitive version of the classical discounted approach is used to obtain convergent approximations to the optimal average cost, and to determine nearly optimal stationary policies. [ABSTRACT FROM AUTHOR]

The major goal of this study is to demonstrate the existence and controllability of a second-order abstract differential system with state-dependent delay. We also discuss the well-posedness of the problem at hand. Finally, some examples are provided for the illustration of the main results. [ABSTRACT FROM AUTHOR]

In this article, new iterative algorithms for solving the discrete Riccati and Lyapunov equations are derived in the case where the transition matrix is diagonalizable with real eigenvalues. It is shown that the proposed iterative algorithms are faster than the classical ones, even for a small number of iterations.

Discrete tempered fractional calculus, as a fresh generalization of discrete fractional calculus, takes both the advantages of discrete fractional calculus and its tempered fractional counterpart, which has great potential value in the interpretation of special physical phenomena. However, its essential characteristics are far from clarifying. Meanwhile, the primary goal of this study is to enrich the vital characteristics of discrete tempered fractional calculus in the presence of required domains for diverse scenarios. First, the reciprocal properties of the tempered fractional summation and difference operators, as well as the discrete version of Taylor's formula, are established. The formulae of the discrete Laplace transform for discrete tempered fractional calculus are also determined with the aid of discrete convolution. Then, the well-posedness of tempered fractional difference equation is investigated, along with the exploration on the corresponding Volterra summation equation, and the existence, uniqueness and continuous dependence as well. In the aspect of application, the criterion of Ulam–Hyers stability of tempered fractional difference equation on finite time interval is obtained in light of a novel discrete Gronwall inequality. Besides, all examples are showed to verify the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

This paper investigates finite-time stability and observer-based finite-time control for nonlinear uncertain switched discrete-time system. Firstly, sufficient conditions are given to ensure that a class of switched nonlinear uncertain discrete-time system is finite-time stable under arbitrary switching. The observer-based controller is constructed. By constructing the switched Lyapunov function, sufficient conditions are derived to ensure the resulting closed-loop system is finite-time stable via observer-based control. The observer-based controller is designed to guarantee a switched nonlinear discrete-time system is finite-time stabilized. Finally, two numerical examples are given to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, we derive formulae for computing the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem with nonconvex objective functions and control constraints. Particularly, we obtain formulae for upper and lower evaluation on the S-derivative of the extremum multifunction via the solution of state equations, the tangent cone to the constraint sets, and the Fréchet derivative of the objective functions. By establishing an abstract result on the S-derivative of the extremum multifunction in a multi-objective parametric mathematical programming problem, we derive formulae for upper and lower evaluation on the S-derivative of the extremum multifunction in a multi-objective parametric discrete optimal control problem. [ABSTRACT FROM AUTHOR]

In this paper, a family of discrete optimal control problems that depend on parameters is considered. The control problems are reformulated as parametric optimization problems. By establishing/exploiting abstract results on subdifferentials of optimal value functions of parametric optimization problems, we derive formulas for estimating/computing subdifferentials of optimal value functions of parametric discrete optimal control problems in both nonconvex and convex cases. Namely, for control problems with nonconvex costs, upper-evaluations on the regular subdifferential and the limiting (Mordukhovich) subdifferential of the optimal value function are obtained without using the (strict) differentiability of the costs. Meanwhile, for control problems with convex costs, besides results on estimating/computing the subdifferential (in the sense of convex analysis) of the optimal value function, it is worth pointing out that some properties of the optimal value function are first discussed in this paper. [ABSTRACT FROM AUTHOR]

This paper studies generalized differentiation for a class of normal cone operators and sensitivity of solution to a class of parametric discrete optimal control problems. We first establish the Fréchet and Mordukhovich coderivatives of a class of normal cone operators in Asplund spaces. We then use the obtained results to compute and estimate the Fréchet coderivative of the solution map to parametric discrete optimal control problems with nonlinear state equations. [ABSTRACT FROM AUTHOR]

This paper deals with the first-order and second-order necessary optimality conditions concerning the components for discrete-time stochastic systems under weakened convexity assumptions. By means of a new discrete-time backward stochastic equation, we establish a first-order necessary optimality condition concerning every component in the form of a global stochastic maximum principle. Besides, we also obtain an optimality condition concerning one component of vector control in the form of a global stochastic maximum principle and concerning the other in the form of linearised stochastic maximum principle. Moreover, by introducing a new discrete-time backward stochastic matrix equation, we derive diverse second-order necessary optimality conditions of singular and quasi-singular controls concerning the components. [ABSTRACT FROM AUTHOR]

In this paper, the exponential synchronization problem of two different discrete-time chaotic neural networks with time delays and stochastic disturbances is investigated. In addition, the unreliable communication links are taken into account between the master system and its slave system, which are modelled as stochastic data dropouts satisfying Bernoulli distributions. By utilizing the Lyapunov functional approach and the stochastic analysis theory, a sufficient condition for the error dynamic system to be mean-square exponentially stable is first obtained. Then based on such sufficient condition, a reliable controller is designed to guarantee that two different discrete-time delayed neural networks with stochastic disturbances are exponentially synchronized in the mean square. The parameters of a desired state feedback controller can be achieved by solving in terms of linear matrix inequality. Finally, a numerical example is presented to validate the feasibility and effectiveness of the proposed synchronization approaches. [ABSTRACT FROM AUTHOR]

The present paper studies the optimization of the Bolza problem with a system of convex and nonconvex, discrete and differential state variable second-order inequality constraints by deriving necessary and sufficient conditions of optimality. The problem with a system of discrete-approximation inequalities is investigated using the proposed method of discretization and equivalence theorems for subdifferential inclusions, which greatly contributes to the derivation of adjoint discrete inclusions generated by a given system of nonlinear inequality constraints. Furthermore, we formulate sufficient conditions of optimality for the continuous problem by passing to the case of limit. A numerical example is provided to illustrate the theoretical approach's effectiveness. [ABSTRACT FROM AUTHOR]

For a discrete dynamical control system with noise, we consider the problem of retaining a phase point in a given family of sets at discrete instants of time. We analyze the case where the control vectogram is a polyhedron defined by a system of linear inequalities. We prove some properties of specific polyhedra satisfying the linearity condition, which allows us to obtain a condition of retention in the explicit form. Necessary and sufficient conditions for the possibility of retention are given. The results obtained are illustrated by an example. [ABSTRACT FROM AUTHOR]

In previous work, the synchronization of multi-links systems with Lévy noise (MLSLN) has not been fully investigated. Therefore, the synchronization of MLSLN is concerned via feedback discrete-time observations control. Therein, we provide a novel Lyapunov functional for MLSLN. Moreover, the upper bound of the state observations interval duration is obtained. By making use of Lyapunov functional method, some techniques of inequality and graph theory, we derive some sufficient conditions to guarantee the mean-squared asymptotical synchronization of MLSLN. Then, the theoretical results are employed to study the synchronization of single-link robot arms. Meantime, numerical simulations are given to demonstrate the effectiveness of the developed results. [ABSTRACT FROM AUTHOR]

This work concerns with semi-Markov decision chains evolving on a finite state space. The controller has a positive and constant risk sensitivity coefficient, and the performance of a control policy is measured by the risk-sensitive average cost criterion. Under conditions ensuring that the optimal value function is determined via a single optimality equation, the fixed points of a family of contractive operators are used to obtain convergent approximations to the optimal average cost and to a solution of optimality equation, extending the classical discounted approach to the context of the paper. In contrast with the Markovian case, the contractive operators utilized in this work depend on two parameters. [ABSTRACT FROM AUTHOR]

We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain. [ABSTRACT FROM AUTHOR]

Nonpharmaceutical interventions (NPI) such as banning public events or instituting lockdowns have been widely applied around the world to control the current COVID-19 pandemic. Typically, this type of intervention is imposed when an epidemiological indicator in a given population exceeds a certain threshold. Then, the nonpharmaceutical intervention is lifted when the levels of the indicator used have decreased sufficiently. What is the best indicator to use? In this paper, we propose a mathematical framework to try to answer this question. More specifically, the proposed framework permits to assess and compare different event-triggered controls based on epidemiological indicators. Our methodology consists of considering some outcomes that are consequences of the nonpharmaceutical interventions that a decision maker aims to make as low as possible. The peak demand for intensive care units (ICU) and the total number of days in lockdown are examples of such outcomes. If an epidemiological indicator is used to trigger the interventions, there is naturally a trade-off between the outcomes that can be seen as a curve parameterized by the trigger threshold to be used. The computation of these curves for a group of indicators then allows the selection of the best indicator the curve of which dominates the curves of the other indicators. This methodology is illustrated with indicators in the context of COVID-19 using deterministic compartmental models in discrete-time, although the framework can be adapted for a larger class of models. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of bounded invariance complexity, bounded invariance complexity in the mean and mean Lyapunov-stability for control systems. Then we characterize these notions by introducing six types of equi-invariability. As a by-product, two new dichotomy theorems for the control system on the control sets are established. [ABSTRACT FROM AUTHOR]

Diabetes is a disease which caused by socio-environmental and / or genetic factors. The negative effect of socio-environmental or lifestyle leads a susceptible individual to become a diabetic. On the one hand, social interaction wields a great deal of influence over lifestyle. On the other hand, genetic factors are the main cause of the birth diabetes genetic disorder. Considering these above mentioned factors. In the present paper, we study a discrete age continuous mathematical model that describes the dynamics of diabetics. We highlight the negative impact of socio-environmental on diabetic patients according to age groups. We also suggest an optimal strategy to implement the best campaigns of rising awareness that aims at protecting diabetic patients from the negative impact of a lifestyle that leads them to complications. In addition to psychological treatment and follow-up of diabetic patients with complications, an awareness campaign will also be carried out for people with potential diabetes that aims at educating them about the dangerous of diabetes and its complications. Pontryagin's maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. The numerical simulation is carried out using MATLAB. [ABSTRACT FROM AUTHOR]

Differential stability of parametric discrete optimal control problems in Banach spaces with nonconvex cost functions, nonlinear state equations and mixed constraints is studied. By establishing an abstract result on the subdifferentials of the value function of a parametric mathematical programming problem with geometrical, equality and inequality constraints, we derive formulas for computing the Mordukhovich subdifferential and the singular subdifferential of the value function to a parametric discrete optimal control problem. [ABSTRACT FROM AUTHOR]

Positive dynamical or control systems have all their variables nonnegative. Euler discretization transforms a continuous-time system into a system on a discrete time scale. Some structural properties of the system may be preserved by discretization, while other may be lost. Four fundamental properties of positive systems are studied in the context of discretization: positivity, positive stability, positive reachability and positive observability. Both linear and nonlinear systems are investigated. [ABSTRACT FROM AUTHOR]

Online customers execute transactions without inspecting products could expect to encounter risks of receiving products with unsatisfactory qualities, especially in food transactions. Thus quality supervision plays a key role in the establishment of trust as well as in the management of risk between online customers and sellers. Most papers in this field are in qualitative nature. In this paper, a quality supervision profit (QSP) model is formulated as a discrete-time optimal control problem. It is a quantitative approach, and it broadens the scope of current research in the area. The quality effort level of online sellers (QELs) and the quality supervision level of the electronic intermediary (QSLm) are considered together with their corresponding profit in the proposed model. The aim is to optimize an overall profit. A case study arising from Suichang's food traceability system (FTS) of farm produce online transaction is carried out in details. The results reveal that QELs, QSLm and the profit distribution coefficient have a strong influence upon the profits of both sides. Finally, some concluding remarks, including potential further research topics, are given. [ABSTRACT FROM AUTHOR]

This paper addresses convergence of iterative learning control for impulsive linear discrete delay systems with randomly varying trial lengths when coefficient matrices of systems are permutable. With the aid of the explicit representation of solutions expressed in discrete matrix delayed exponential, we provide two sufficient conditions of convergence to guarantee that tracking errors uniformly converge to zero in the sense of expectation for the above impulsive controlled systems by designing two proper update learning laws with the modified tracking errors. Finally, two illustrative examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper discusses differential stability of convex discrete optimal control problems with mixed constraints. Among the obtained results, we give formulas for exact computing the subdifferential and the singular subdifferential of the optimal value function via suitable multiplier sets. [ABSTRACT FROM AUTHOR]

This paper is concerned with a dual object of the so-called viability kernel of a control system with mixed (state-control) constraints. This object, called the set of sustainable thresholds, describes the possible thresholds for which a given initial condition is viable. In this work, we are concerned with characterizing the weak and strong Pareto fronts of the set of sustainable thresholds and providing a practical method for computing such objects based on optimal control theory and a level-set approach. A numerical example, relying on renewable resource management, is shown to demonstrate the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability for discrete-time Takagi and Sugeno (T-S) fuzzy systems with perturbation disturbance and time delay. Stochastic delay-dependent stability criteria are derived for stochastic T-S fuzzy systems with time-invariant and time-varying delays, respectively. For the time-varying delay case, a novel fuzzy Lyapunov–Krasovskii functional (LKF) without requiring all the involved symmetric matrices to be positive definite is constructed to reduce the conservatism. These stability conditions are then represented in terms of finite linear matrix inequalities (LMIs), which can be solved efficiently by using standard LMIS optimisation techniques. Two numerical examples are given to illustrate the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Incremental stability and convergence properties for forced, infinite-dimensional, discrete-time Lur'e systems are addressed. Lur'e systems have a linear and nonlinear component and arise as the feedback interconnection of a linear control system and a static nonlinearity. Discrete-time Lur'e systems arise in, for example, sampled-data control and integro-difference models. We provide conditions, reminiscent of classical absolute stability criteria, which are sufficient for a range of incremental stability properties and input-to-state stability (ISS). Consequences of our results include sufficient conditions for the converging-input converging-state (CICS) property, and convergence to periodic solutions under periodic forcing. [ABSTRACT FROM AUTHOR]

This work concerns with discrete-time Markov decision processes on a denumerable state space. Assuming that the decision maker is risk-averse with constant risk-sensitivity coefficient, the performance of a control policy is measured by an average criterion associated with a non-negative and bounded cost function. Under conditions ensuring that the optimal average cost is constant, but not necessarily determined via the average cost optimality equation, it is shown that a discounted criterion can be used to approximate the optimal average index. [ABSTRACT FROM AUTHOR]