Stochastic Evolution Systems Research Articles

Linear quadratic optimal control problems of infinite‐dimensional mean‐field type with jumps

AbstractIn this paper, we formulate and investigate a framework for the theory of the linear quadratic optimal control problem (LQ problem) for infinite‐dimensional mean‐field stochastic evolution systems with jumps. We ensure the well‐posedness of the investigated problems by establishing the existence, uniqueness, and a priori estimates for mild solutions to general infinite‐dimensional mean‐field forward stochastic evolution equations (MF‐SEE) and mean‐field backward stochastic evolution equations (MF‐BSEE) with jumps. Leveraging the Yosida approximation theory, we establish a dual theory between MF‐SEE and MF‐BSEE with jumps, overcoming the challenge posed by the inapplicability of Itô's formula in the context of mild solutions. Our main results regarding the existence and uniqueness of the optimal control, along with corresponding dual characterizations and state feedback representations, are obtained through convex analysis techniques, our established dual theory, and decoupling methods.

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

This study focuses on optimal feedback control for stochastic Caputo fractional evolution systems in separable Hilbert spaces. Initially, we deal with the existence of a mild solution, establishing it through the application of the generalised Clarke's subdifferential and a fixed point theorem for multivalued maps. Then, we utilise the Cesari property and the Filippov theorem to demonstrate the existence of feasible pairs. Additionally, we provide pairs of suitable feedback controls under specific conditions. Finally, we present a simple illustration to support our theoretical findings.

An analysis on the approximate controllability outcomes for fractional stochastic Sobolev‐type hemivariational inequalities of order 1<r<2$$ 1<r<2 $$ using sectorial operators

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev‐type hemivariational differential systems of order with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed‐point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

Approximate controllability of second-order neutral stochastic differential evolution systems with random impulsive effect and state-dependent delay

<p>In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.</p>

Approximate Controllability Outcomes of Impulsive Second-Order Stochastic Neutral Differential Evolution Systems

In this manuscript, we present a collection of suitable requirements for the approximate controllability outcomes of impulsive second-order stochastic neutral differential evolution systems. We use the ideas from the sine and cosine functions and the fixed point strategy to demonstrate the primary findings. The analysis then moves to second-order nonlocal stochastic neutral differential systems. Eventually, in order to make our topic more useful, we will provide an epistemological implementation.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Hilfer fractional neutral stochastic Sobolev-type evolution hemivariational inequality: Existence and controllability☆

This paper discusses the approximate controllability of hemivariational inequalities of the Sobolev-type Hilfer fractional neutral stochastic evolution system. Firstly, by using fixed point theorems, fractional calculus, Hilfer fractional derivatives, multivalued maps and stochastic analysis, for stochastic fractional evolution systems, we demonstrate mild solutions. Whereupon we introduce necessary conditions that ensure the approximate controllability of stochastic fractional systems. As a final step, we define our primary outcomes using an example.

Mean Square Finite-Approximate Controllability of Semilinear Stochastic Differential Equations with Non-Lipschitz Coefficients

In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered.

EXISTENCE AND CONTROLLABILITY FOR IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION SYSTEMS WITH STATE-DEPENDENT DELAY

This paper is concerned with the impulsive fractional stochastic neutral evolution systems with state-dependent delay and nonlocal condition. First, the existence of solutions of considered evolution systems are obtained by applying the Banach contraction theorem. Then, on the basis of existence of solutions, the controllability concept of the system is investigated. The main aim is to derive some conditions that could be applied to analyze the controllability results for the considered evolution systems involving state-dependent delay. Finally, the efficiency of theoretical analysis is verified by an example.

First order necessary condition for stochastic evolution control systems with random generators

<p style='text-indent:20px;'>The main purpose of this paper is to establish the first order necessary optimality condition for optimal control problems of stochastic evolution systems with random generators. For our controlled system, both the drift and the diffusion terms contain the control variable, and the control region is a nonempty closed subset of a separable Hilbert space. Compared with the existing works, the main difficulties here are to prove the well-posedness of the control and the adjoint systems. We obtain the well-posedness of the adjoint system in the sense of mild solutions and that of the control system by means of the stochastic transposition method. In addition, the variational analysis approach is employed to handle the nonconvexity of the control region when deriving our optimality condition.</p>

New discussion regarding approximate controllability for Sobolev-type fractional stochastic hemivariational inequalities of order [formula omitted

In this paper, we deal with the approximate controllability of Sobolev-type fractional stochastic evolution hemivariational inequalities of order r∈(1,2). Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, and fixed point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

Approximate Controllability of Non-Instantaneous Impulsive Stochastic Evolution Systems Driven by Fractional Brownian Motion with Hurst Parameter H∈(0,12)

This paper initiates a study on the existence and approximate controllability for a type of non-instantaneous impulsive stochastic evolution equation (ISEE) excited by fractional Brownian motion (fBm) with Hurst index 0<H<1/2. First, to overcome the irregular or singular properties of fBm with Hurst parameter 0<H<1/2, we define a new type of control function. Then, by virtue of the stochastic analysis theory, inequality technique, the semigroup approach, Krasnoselskii’s fixed-point theorem and Schaefer’s fixed-point theorem, we derive two new sets of sufficient conditions for the existence and approximate controllability of the concerned system. In the end, a concrete example is worked out to demonstrate the applicability of our obtained results.

Construction controllability for conformable fractional stochastic evolution system with noninstantaneous impulse and nonlocal condition

Conformable fractional stochastic differential equation with noninstantaneous impulse and Poisson jump via nonlocal condition is studied. Controllability for the considered problem is constructed. The required results are established based on fractional calculus, stochastic analysis, and Sadovskii’s fixed point theorem. Moreover, an example is provided to illustrate the applicability of the results.

Well-posedness for the coupling of a random heat equation with a multiplicative stochastic Barenblatt equation

In this contribution, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a random heat equation coupled with a Barenblatt’s type equation with a multiplicative stochastic force in the sense of Itô. In a first step, we establish well-posedness in the case of an additive noise through a semi-implicit time discretization of the system. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case.

Stochastic Optimal Control Problem in Advertising Model with Delay

This paper investigates the optimal control problem arising in advertising model with delay. The authors reformulate the problem in Hilbert space by stochastic evolution equation and consider the optimal control problem of controlled stochastic evolution system. The necessary and sufficient optimality conditions of the control are established. The proposed approach is different from most existing studies of optimal advertising policy problem with delay. These results are applied to the optimal advertising policy problem under two different structures and the optimal advertising strategies are obtained.

Asymptotic Dissipativity for Merged Stochastic Evolutionary Systems with Markov Switchings and Impulse Perturbations under Conditions of Lévy Approximation

Conditions for asymptotic dissipativity are established for the merged system of stochastic differential equations with Markov switchings and impulse perturbations under conditions of Levy approximation. In particular, it is analyzed how the behavior of the boundary process depends on the pre-limiting normalization of a stochastic evolution system in the ergodic Markovian environment under the conditions of Levy approximation.

Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness &#x0028 existence and uniqueness of the strong variational solution &#x0029 and asymptotic stability in probability of nonlinear stochastic evolution systems &#x0028 SESs &#x0029 driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation &#x0028 HJBE &#x0029. An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.

Uniform Convergence of the Solutions of Riccati Equations for a Family of Optimal Control Problems

The uniform convergence of the solutions of a parameterized family of Riccati differential equations which arise in the context of optimal control problems for systems described by a family of linear time-varying evolution equations is considered on the infinite time interval in Hilbert space. Some sufficient conditions and an illustrative example are given for uniform convergence of the solutions of a family of the Riccati differential equations. The uniform convergence of the solutions of the Riccati differential equations with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.

On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity

The limiting behavior of stochastic evolution processes with small noise intensity ϵ is investigated in distribution-based approaches. Let μϵ be a stationary measure for stochastic process Xϵ with small ϵ and X0 be a semiflow on a Polish space. Assume that {μϵ: 0 < ϵ ⩽ ϵ0} is tight. Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or Levy processes.

Optimal control of a class of semi‐linear stochastic evolution equations with applications

In this study, the authors derive a new type of maximum principle for both stochastic evolution and parabolic type stochastic partial differential systems. The systems only require the Hamiltonian functional to be concave in the state variable rather than in both state and control variables. They also show a connection between these two types of systems. Finally, examples are given to illustrate the authors' theoretical results.