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In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear ψ-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index H∈(1/2,1). At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results. [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]

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 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]

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]

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]