Search Results (4)

Controllability is a basic problem in the study of stochastic generalized systems. Compared with ordinary stochastic systems, the structure of stochastic singular systems is more complex, and it is necessary to study the controllability of stochastic generalized systems in the context of different solutions. In this paper, the controllability of semilinear stochastic generalized systems was investigated by using a GE-evolution operator for integral and impulsive solutions in Hilbert spaces. Some sufficient and necessary conditions were obtained. Firstly, the existence and uniqueness of the integral solution of semilinear stochastic generalized systems were discussed using the GE-evolution operator theory and Banach fixed point theorem. The existence and uniqueness theorem of the integral solution was obtained. Secondly, the approximate controllability of semilinear stochastic generalized systems was studied in the case of the integral solution. Thirdly, the existence and uniqueness of the impulsive solution of semilinear stochastic generalized systems were considered, and some sufficient conditions were provided. Fourthly, the approximate controllability of semilinear stochastic generalized systems was studied for the impulsive solution. At last, the exact controllability of linear stochastic systems was studied in the case of the impulsive solution, with some necessary and sufficient conditions given. The obtained results have important theoretical and practical value for the study of controllability of semilinear stochastic generalized systems. Full article

A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems. Full article

Approximate controllability of two types of nonlinear stochastic generalized systems is investigated in the sense of mild solution in Hilbert spaces. Firstly, the approximate controllability of semilinear stochastic generalized systems with control only acting on the drift terms is discussed by GE-evolution operator and Nussbaum fixed-point theorem. Secondly, the approximate controllability of semilinear stochastic systems with control acting on both drift and diffusion terms is handled by using GE-evolution operator and Banach fixed-point theorem. At last, two illustrative examples are given. Full article

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research. Full article