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This paper is devoted to the problem of indefinite stochastic linear quadratic (LQ) optimal control by piecewise constant controls in an infinite horizon case. By restricting the set of admissible controls to the class of piecewise constant stochastic processes, we reformulated the above control problem under the setting of systems modeled by Itô differential equations controlled by impulses. We show that the solution in a state feedback form of the indefinite stochastic LQ control problem is equivalent to the existence of a global stabilizing solution associated to a class of backward matrix linear differential equations with a Riccati type jumping operators. [ABSTRACT FROM AUTHOR]

We investigate a linear quadratic tracking problem for impulsive controlled stochastic systems. The main tool in the derivation of the optimal control is the stabilizing solution for a class of backward jump matrix linear differential equation. We introduce the concept of mean square stability by impulses and consider the properties of stabilizing solution of the associated backward jump matrix linear differential equation with Riccati‐type operators. Necessary and sufficient conditions that guarantee the existence of the unique optimal control for the linear quadratic tracking problem for impulsive controlled stochastic systems are proved. [ABSTRACT FROM AUTHOR]

We investigate a problem to solve the linear quadratic tracking problem for stochastic systems controlled by impulses. Two optimal control problems are investigated where the different objective functions are minimized. Explicit formulae for optimal controls are developed. The optimal controllers are computed based on the solution of the backward jump matrix Lyapunov type linear differential equations. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the exponential stability of nonsingular impulsive delayed integro‐differential systems. Our approach to addressing nonsingular differential problems involves the application of permutable matrices and their associated delayed exponential. Furthermore, we investigate the controllability of a nonlinear impulsive and delayed problem by employing the corresponding Gramian matrix. Finally, to illustrate the theoretical outcomes, we provide examples and graphical representations for each situation. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to formulating exponential behaviors like stability/instability for the linear time-varying systems and for the adjoint one. The classical concept of uniform exponential stability is generalized. Using this generalized concepts, some results extending existing uniform exponential stability conditions for linear time-varying systems are derived. As special cases for these results, some conditions are derived for the adjoint system. A characterization of the generalized concepts in terms of Lyapunov sequences is also given. Also, an example is included to further illustrate the connection with the classical concept of uniform exponential stability. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

The concept of generalized exponential trichotomy for linear time-varying systems is investigated in relationship with the classical notion of uniform exponential trichotomy. Some key properties of generalized exponential trichotomy are explored through supplementary projections. These results are also extended to the case of projection sequences, while certain applications for adjoint systems are suggested. [ABSTRACT FROM AUTHOR]

An optimal control problem for a system of linear Itô differential equations with two fast time scales and Markovian jumping is considered. The asymptotic structure for the stabilizing solution satisfying a sign condition for the coupled stochastic algebraic Riccati equations is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a unique solution is established for a coupled system of Langevin fractional problems of ψ -Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulses using the Banach contraction principle. We also find at least one solution to the aforementioned system using some assumptions and Schaefer's fixed point theorem. After that, Ulam–Hyers stability is discussed. Finally, to provide additional support for the main results, pertinent examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and uniqueness along with Ulam‐type stability of fractional integro causal evolution impulsive systems on time scales. To establish our main results, we use Picard operator along with abstract Grönwall inequality and Gröwall lemma. We constitute some assertions to remove difficulties. At the end, the results are demonstrated via the support of an example. [ABSTRACT FROM AUTHOR]

This article investigates the exact and approximate controllability, finite time stability, and B$$ \mathcal{B} $$–Hyers–Ulam–Rassias stability of a neutral impulsive system of differential equations with two delays. Fixed point theory is associated in the analysis of controllability of the system. The stability analysis is carried out with the help of Grönwall type inequality. [ABSTRACT FROM AUTHOR]

In this paper we consider some concepts of exponential splitting for nonautonomous linear discrete-time systems. These concepts are generalizations of some well-known concepts of (uniform and nonuniform) exponential dichotomies. Connections between these concepts are presented and some illustrating examples prove that these are distinct. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present necessary and sufficient conditions for nonuniform power instability property of linear discrete-time systems in Banach spaces. A characterization of the nonuniform power instability in terms of Lyapunov sequences is also given. [ABSTRACT FROM AUTHOR]

In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate four concepts of exponential stability for difference equations in Banach spaces. Characterizations of these concepts are given. They can be considered as variants for the discrete-time case of the classical results due to Barbashin and Datko . An illustrative example clarifies the relations between these concepts. [Copyright &y& Elsevier]

This paper considers nonuniform exponential stability and nonuniform exponential instability concepts for one-sided discrete-time random dynamical systems. These concepts are generalizations from the deterministic case. Using this, characterizations in terms of Lyapunov functions respectively Lyapunov norms are presented. Also, an approach in terms of considered concepts for the inverse and adjoint random discrete-time systems is derived. [ABSTRACT FROM AUTHOR]

In this study, we investigate the nonuniform exponential stability and nonuniform exponential instability concepts for a linear random one‐sided system. We derive necessary and sufficient conditions for these concepts. As an extension for this study, we establish Lyapunov‐type characterizations using Lyapunov functions and Lyapunov norms. Additionally, we illustrate the essential properties that are derived in these directions for the adjoint and inverse systems associated for the original linear random one‐sided system. [ABSTRACT FROM AUTHOR]

In this dissertation, we study the coupled system of q$$ q $$‐fractional Langevin differential equations involving q$$ q $$‐Caputo derivative having q$$ q $$‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we examine a coupled system of fractional integrodifferential equations of Liouville–Caputo form with instantaneous impulsive conditions in a Banach space. We obtain the existence and uniqueness results by applying the theory of fixed point theorems. In a similar manner, we discuss Hyers–Ulam stability and controllability. We also present an example to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

A stability problem for a class of large-scale singularly perturbed stochastic systems (SPSSs) with state-multiplicative white noise and Markovian jumping parameters is considered. Based on the linear evolution operator, an exponential stability in mean square is investigated. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of optimal H 2 filtering for a class of continuous-time stochastic systems without assuming their exponential stability in the mean square sense. Indeed, the stability assumption is relaxed and we assume instead that the Lyapunov operator associated to the dynamical stochastic system is exponentially dichotomic . The optimal solution of the considered optimization problem is expressed in terms of the unique solution of a suitable algebraic Lyapunov equation and the stabilizing solution of a certain algebraic Riccati equation. [ABSTRACT FROM AUTHOR]

Horadam sequence is a general recurrence of second order in the complex plane, depending on four complex parameters (two initial values and two recurrence coefficients). These sequences have been investigated over more than 60 years, but new properties and applications are still being discovered. Small parameter variations may dramatically impact the sequence orbits, generating numerous patterns: periodic, convergent, divergent, or dense within one dimensional curves. Here we explore Horadam sequences whose orbit is dense within a 2D region of the complex plane, while the complex argument is uniformly distributed in an interval. This enables the design of a pseudo-random number generator (PRNG) for the uniform distribution, for which we test periodicity, correlation, Monte Carlo estimation of π , and the NIST battery of tests. We then calculate the probability distribution of the radii of the sequence terms of Horadam sequences. Finally, we propose extensions of these results for generalized Horadam sequences of third order. [ABSTRACT FROM AUTHOR]

This paper is devoted to the application of the recently developed results about exact detectability of a large class of time-varying hybrid stochastic systems (Drăgan et al., 2020). More specifically, we will show from one side the connection between this new detectability notion and the derivation of a criterion for exponential stability in mean square for this class of systems. In the second part, we will show the central role played by exact detectability in Riccati equations arising in linear quadratic (LQ) regulator problems for such stochastic systems: it is a necessary and sufficient condition for the Riccati equation to have a unique positive semi-definite and stabilizing solution. [ABSTRACT FROM AUTHOR]

In this paper, we examine a sampled-data Nash equilibrium strategy for a stochastic linear quadratic (LQ) differential game, in which admissible strategies are assumed to be constant on the interval between consecutive measurements. Our solution first involves transforming the problem into a linear stochastic system with finite jumps. This allows us to obtain necessary and sufficient conditions assuring the existence of a sampled-data Nash equilibrium strategy, extending earlier results to a general context with more than two players. Furthermore, we provide a numerical algorithm for calculating the feedback matrices of the Nash equilibrium strategies. Finally, we illustrate the effectiveness of the proposed algorithm by two numerical examples. As both situations highlight a stabilization effect, this confirms the efficiency of our approach. [ABSTRACT FROM AUTHOR]

This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in Banach spaces. Variants for weak exponential stability of some well-known results in uniform stability theory (Buşe and Niculescu (2009) , Daleckij and Krein (1974) , Datko (1973) , Rolewicz (1986) , Stoica and Megan (2009) ) are obtained. [ABSTRACT FROM AUTHOR]

The problem of sampled-data Nash equilibrium strategy in a state feedback setting for a stochastic linear quadratic differential game is addressed. It is assumed that the admissible strategies are constant on the interval between two measurements. The original problem is converted into an equivalent one for a linear stochastic system with finite jumps. This new formulation of the problem allows us to derive necessary and sufficient conditions for the existence of a sampled-data Nash equilibrium strategy in a state feedback form. These conditions are expressed in terms of solvability of a system of interconnected matrix linear differential equations with finite jumps and subject to some algebraic constraints. We provide explicit formulae of the gain matrices of the Nash equilibrium strategy in the class of piece-wise constant strategies in a state feedback form. The gain matrices of the feedback Nash equilibrium strategy are computed based on the solution of the considered system of matrix linear differential equations with finite jumps. For the implementation of these strategies only measurements at discrete-time instances of the states of the dynamical system are required. Finally, we show that under some additional assumptions regarding the sign of the weights matrices in the performance criteria of the two players, there exists a unique piecewise Nash equilibrium strategy in a state feedback form if the maximal length of the sampling period is sufficiently small. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann–Liouville fractional integral boundary conditions. We establish some existence and uniqueness results for the given problem by applying the tools of fixed point theory. Furthermore, we investigate different kinds of stability such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we give two examples to demonstrate the validity of main results. [ABSTRACT FROM AUTHOR]

This primer article focuses on the representation of solutions and finite-time stability of impulsive first-order delay differential systems. We define delayed matrix function with impulses and use variation of parameters to obtain a representation of solutions of linear systems with impulse effects. The famous classical Grownwall inequalities and properties of delayed matrix exponential with impulses are used to develop sufficient conditions for finite-time stability. In the end, we provide some examples to support the results. [ABSTRACT FROM AUTHOR]

We study a coupled system of implicit differential equations with fractional-order differential boundary conditions and the Riemann–Liouville derivative. The existence, uniqueness, and at least one solution are established by applying the Banach contraction and Leray–Schauder fixed point theorem. Furthermore, Hyers–Ulam type stabilities are discussed. An example is presented to illustrate our main result. The suggested system is the generalization of fourth-order ordinary differential equations with anti-periodic, classical, and initial boundary conditions. [ABSTRACT FROM AUTHOR]