Stochastic Parabolic Systems Research Articles

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

On Some Stochastic Parabolic Systems Driven by New Fractional Brownian Motions

On Some Stochastic Parabolic Systems Driven by New Fractional Brownian Motions

Editor’s Note: “Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach” [Nonlinear Analysis: Real World Applications, Volume 12, Issue 3 (2011) 1386-1400

Editor’s Note: “Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach” [Nonlinear Analysis: Real World Applications, Volume 12, Issue 3 (2011) 1386-1400

A mobile sensing approach to distributed consensus filtering of 2D stochastic nonlinear parabolic systems with disturbances

The focus of this paper is on distributed consensus filtering in two-dimensional (2D) space for stochastic nonlinear parabolic systems with disturbances. Distributed filters are used in mobile sensor networks to achieve consensus estimate. By utilizing a mobile sensing approach, the optimal framework for parabolic systems to improve filtering performance is provided. Sufficient conditions are created under the filtering error is boundedness by employing operator-dependent Lyapunov functional. The velocity law of each mobile sensor can also be used to guide the filtering error system toward rapid convergence. Finally, numerical examples are used to verify the effectiveness of the suggested approach.

Influence of numerical discretizations on hitting probabilities for linear stochastic parabolic systems

This paper investigates the influence of standard numerical discretizations on hitting probabilities for a system of linear stochastic parabolic equations driven by space-time white noises. We establish lower and upper bounds for hitting probabilities of the associated numerical solutions of both temporal and spatial semi-discretizations in terms of Bessel–Riesz capacity and Hausdorff measure, respectively. Moreover, the critical dimensions of both temporal and spatial semi-discretizations turn out to be halves of those of the exact solution. This reveals that there exist some Borel sets A such that the probability of the event that the paths of the numerical solution hit A cannot converge to that of the exact solution when the stepsizes vanish.

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.

Controllability and observability of some coupled stochastic parabolic systems

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.

EXISTENCE RESULTS FOR A NEW CLASS OF FRACTIONAL IMPULSIVE PARTIAL NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, a new class of fractional impulsive partial neutral stochastic integro-differential equations with infinite delay is introduced.Under some dissipative conditions, we obtain the existence, uniqueness and continuous dependence of mild solutions for these equations. An application involving a fractional stochastic parabolic system with not instantaneous impulses is considered.

Global Carleman estimate for stochastic parabolic equations, and its application

This paper is addressed to proving a new Carleman estimate for stochastic parabolic equations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X. Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm. 5.2), one extra gradient term involving in that estimate is eliminated. Also, our improved Carleman estimate is established by virtue of the known Carleman estimate for deterministic parabolic equations. As its application, we prove the existence of insensitizing controls for backward stochastic parabolic equations. As usual, this insensitizing control problem can be reduced to a partial controllability problem for a suitable cascade system governed by a backward and a forward stochastic parabolic equation. In order to solve the latter controllability problem, we need to use our improved Carleman estimate to establish a suitable observability inequality for some linear cascade stochastic parabolic system, while the known Carleman estimate for forward stochastic parabolic equations seems not enough to derive the desired inequality.

Controllability of Some Coupled Stochastic Parabolic Systems with Fractional Order Spatial Differential Operators by One Control in the Drift

This paper is addressed to a study of the null/approximate controllability for a class of coupled systems governed by two linear forward stochastic parabolic equations with fractional order spatial differential operators. Our method is based on the Lebeau--Robbiano strategy. The key is to establish a suitable observability estimate for some coupled fractional order backward stochastic parabolic systems with terminal states in finite dimensional spaces. Compared to deterministic coupled parabolic systems, the coupling appearing in diffusion terms in the stochastic case introduces quite interesting new phenomena. We present a somewhat surprising counterexample to show that the controllability of coupled stochastic parabolic systems is not robust with respect to the coupling coefficient in diffusion terms. This indicates that the usual Carleman-type estimate approach does not seem to work for our controllability problem. Moreover, our controllability results for parabolic systems with fractional order spatia...

Practical stability of nonlinear stochastic hybrid parabolic systems of Itô-type: Vector Lyapunov functions approach

Lyapunov’s second method is widely recognized as a fundamental tool not only in the theory of stability but also in studying other qualitative properties of solutions of differential equations. The main characteristic of this method is the utilization of a function, namely the Lyapunov function, together with the theory of differential and integral inequalities to develop a very general comparison principle under much less restrictive conditions to investigate qualitative and quantitative properties of nonlinear differential equations. The most important advantage of this technique is that it does not require the explicit knowledge of solutions. This paper is devoted to extend this technique to investigate the practical stability of the solution process of stochastic hybrid parabolic partial differential equations of Itô type under Markovian structural perturbations. The jump nonlinear system is a hybrid system with state vector that has two components x ( t ) and η ( t ) . Here x ( t ) is referred to as the state and η ( t ) is referred as the mode. During the operation, the system can jump from one mode to another in a random way, which makes this class of systems a stochastic one. The switching between the modes is governed by a Markov process with discrete and finite state space. When the system mode is fixed it evolves like a deterministic nonlinear system. This kind of system can be used to describe abrupt phenomena, such as component and interconnection failures.

STOCHASTIC PARABOLIC MODEL FOR INFINITE-DIMENSIONAL FORWOARD RATE AND MEAN-VARIANCE OPTIMAL CONTROL

We consider the term structure modeling by using a appropriate stochastic parabolic system with boundary noises. After finding a sufficient condition for the no arbitrage opportunity, we solve the mean-variance optimal control problem in the incomplete market.

On maximum likelihood identification of spatially-varying parameters in stochastic parabolic systems

The maximum likelihood estimation (M.L.E.) for spatially-varying parameters in stochastic parabolic systems is studied. The main result is to show the consistency property of the M.L.E. for unknown parameters by using the method of sieves, i.e., first the admissible class of unknown parameters is projected into a finite-dimensional space and next the convergence of the derived finite-dimensional M.L.E. to the infinite-dimensional M.L.E. is justified under some conditions. Finally numerical examples are presented.

State Estimation for Stochastic Parabolic Systems with Unknown Boundary Condition

State Estimation for Stochastic Parabolic Systems with Unknown Boundary Condition

Identification of a Discontinuous Parameter in Stochastic Parabolic Systems

The purpose of this paper is to study the identification problem for a spatially varying discontinuous parameter in stochastic diffusion equations. The consistency property of the maximum likelihood estimate (M.L.E.) and a generating algorithm for M.L.E. have been explored under the condition that the unknown parameter is in a sufficiently regular space with respect to spatial variables. In order to prove the consistency property of the M.L.E. for a discontinuous diffusion coefficient, we use the method of sieves, i.e., first the admissible class of unknown parameters is projected into a finite-dimensional space and next the convergence of the derived finite-dimensional M.L.E. to the infinite-dimensional M.L.E. is justified under some conditions. An iterative algorithm for generating the M.L.E. is also proposed with two numerical examples.

On adaptive boundary control for stochastic parabolic systems with unknown potential coefficient

An adaptive boundary control problem for a stochastic heat diffusion equation is studied. The considered system contains an unknown potential coefficient which is a function of the spatial variables. The estimation algorithm for the unknown potential coefficient is proposed by using the stochastic approximation technique. After showing the strong consistency of the estimated parameter, the cost for the adaptive control scheme presented here is shown to converge to the optimal ergodic cost. Finally some numerical examples are shown.

Weak formulation of stochastic parabolic systems with subdifferential boundary conditions

AbstractThis paper discusses the mathematical formulation of the stochastic parabolic system, which is described by the multivalued operator called subdifferential operator, that plays an important role in the modeling of the free boundary problem such as viscoelastic and obstacle problems. For various deterministic distributed systems containing a subdifferential operator, the existence of the strong or weak solution is already shown.With respect to the stochastic distributed system, on the other hand, the existence of a strong solution has not been shown except for the system containing a subdifferential operator of a special convex function. As to the existence of the weak solution, there is a report of the case where the subdifferential operator appears in the interior of the system, but there is no report on the general stochastic system containing a subdifferential operator in the boundary condition. Another point is that a strong constraint is imposed on the initial value and the input function in the proof for the existence of the strong solution.From those viewpoints, this paper aims at ameliorating such a constraint and formulates the system by the weak solution. First, the existence condition for the weak solution is shown. Then it is shown that there exists the maximal solution in the set of weak solutions and an estimate for the maximal solution is derived which has a stronger significance than in the past formulation.

On the Weak Formulation of Stochastic Parabolic Systems with Boundary Conditions of Subdifferential Type

On the Weak Formulation of Stochastic Parabolic Systems with Boundary Conditions of Subdifferential Type

On the Lagrange and Euler approaches to the construction of solutions of stochastic semilinear parabolic systems

On the Lagrange and Euler approaches to the construction of solutions of stochastic semilinear parabolic systems

Algebraic Riccati Equation Arising in Boundary Control Problems

An algebraic Riccati equation is studied, with application to the optimal control of deterministic and stochastic parabolic systems with boundary control. The control function can act on the boundary through Dirichlet or Neumann conditions.