Systems Of Parabolic Type Research Articles

Minimization of integral quadratic estimate of controlled variable in systems with distributed parameters

A constructive method for solving the linear-quadratic problem of optimal control of a parabolic-type system with distributed parameters is proposed under the condition of uniform estimation of target sets. The optimality criterion takes the form of an integral quadratic estimate of the controlled state function in the spatio-temporal domain of its definition. A parameterized representation of control inputs is given with the required accuracy within special intervals of the optimal process, where control inputs cannot be determined using first-order analytical optimality conditions. The suggested approach is based on a previously developed alternance method for constructing parameterized algorithms of programmed control, which heavily relies on fundamental regularities of the subject area. It is demonstrated that the equations of the optimal regulators within the special intervals are reduced to the linear feedback algorithms based on the measured states of the objects. These algorithms are supplemented with switches at boundary points to apply admissible control inputs corresponding to the calculated values of the controlled variable.

Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous/Homogeneous Cauchy-Neumann/Neumann Boundary Conditions

Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class Wp1,2(Q)×Wν1,2(Q). One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, alg-frac_sec-ord-varphi_PHT, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper.

Bounded on the semi-axis multiperiodic solution of a linear finite-hereditarity integro-differential equation of parabolic type

The question of the existence of a solution of linear integro-differential systems of parabolic type limited on the semiaxis in a spatial variable and multiperiodic in time variables was considered. Sufficient conditions of multiperiodic oscillations in time variables in a linear homogeneous equation with a boundary condition and in a linear inhomogeneous equation were established. A linear homogeneous and inhomogeneous finitehereditarity integro-differential equation of convective-diffusion type were investigated.

Boundedness of the solutions of a kind of nonlinear parabolic systems

We deal with nonlinear systems of parabolic type satisfying componentwise structural conditions. The nonlinear terms are Carathéodory maps having controlled growth with respect to the solution and the gradient and the data are in anisotropic Lebesgue spaces. Under these assumptions we obtain essential boundedness of the weak solutions.

TheH∞-control problem for parabolic systems. Applications to systems with singular Hardy potentials

We solve theH∞-control problem with state feedback for infinite dimensional boundary control systems of parabolic type with distributed disturbances and provide some applications of this result to equations with Hardy potentials with the singularity inside or on the boundary, in the cases of a distributed control and of a boundary control.

On a moving boundary problem associated with the swelling drug release platforms

This paper studies a parabolic-type system of time-dependent partial differential equations coupled with Robin boundary conditions to model the solvent penetration and subsequently a controlled drug release from a swellable polymeric spherical platform. The model's key feature is that the solvent penetration and the polymer swelling result in moving boundaries. From an analytical perspective, some properties of the solutions, such as the positivity and the uniqueness of the solutions, are established. A second-order iterative procedure based on the backward finite difference method is carried out to solve the problem numerically. Finally, the numerical results and the asymptotic solutions are compared, intending to illustrate the validity and applicability of the numerical scheme.

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions u↦fl(t,x,u),l=1,2 and no compact property of C0-semigroups eAl(t),l=1,2.

Non-well-ordered lower and upper solutions for semilinear systems of PDEs

We prove existence results for systems of boundary value problems involving elliptic second-order differential operators. The assumptions involve lower and upper solutions, which may be either well-ordered, or not at all. The results are stated in an abstract framework, and can be translated also for systems of parabolic type.

Funnel control for the monodomain equations with the FitzHugh-Nagumo model

We consider a nonlinear reaction diffusion system of parabolic type known as the monodomain equations, which model the interaction of the electric current in a cell. Together with the FitzHugh-Nagumo model for the nonlinearity they represent defibrillation processes of the human heart. We study a fairly general type with co-located inputs and outputs describing both boundary and distributed control and observation. The control objective is output trajectory tracking with prescribed performance. To achieve this we employ the funnel controller, which is model-free and of low complexity. The controller introduces a nonlinear and time-varying term in the closed-loop system, for which we prove existence and uniqueness of solutions. Additionally, exploiting the parabolic nature of the problem, we obtain Hölder continuity of the state, inputs and outputs. We illustrate our results by a simulation of a standard test example for the termination of reentry waves.

Adaptive H∞ Consensus Control for Distributed Parameter Systems of Parabolic Type on Directed Network Graph

A design method of adaptive H ∞ consensus control of multi-agent systems composed of a class of distributed parameter systems (infinite-dimensional systems) of parabolic type on directed network graphs, is presented in this paper. The proposed control scheme is formulated as a solution of certain H ∞ control problems, where the effects of neglected infinite-dimensional modes are regarded as external disturbances to the process. It is shown that the resulting control system is robust to uncertain system parameters and that desirable consensus tracking is achieved approximately via finite dimensional adaptive controllers.

On the bicomponent contaminant transport in wetland flow with reactions

Contaminant transport in the wetland flow is subject to various reactions. Most researches on this topic have limited to the single component contaminant, relatively little has concerned the effect of competitive reaction on the transport of coupled bicomponent contaminant. This work analytically studies the bicomponent contaminant transport in the free surface wetland flow under the combined conditions of reversible competitive transfer, irreversible degradation and bed absorption, by solving the constant-coefficient second-order linear system of parabolic type and the method of concentration moments. Up to the fourth order concentration moment in the previous work (Wang and Chen, 2017a) is applied to support the fourth order expansion of Hermite polynomials to rigorously derive the solutions with high accuracy. The results demonstrate that the masses of binary components decay at different rate, and the vertical concentration distributions of binary components are tremendously non-uniform in the asymptotic dispersion stage. Three types of reactions in addition to the hydraulic dispersion exert separate control on the concentration distributions. It suggests that the peak concentration rather than the mean should be based for strict environmental implements.

Effects of Straw and Biogas Co-firing on the Combustion Characteristics

In this study, effects of straw co-firing with a renewable fuel – low heating value biogas – on the combustion characteristics, heat output from the device and composition of flue gas, are studied and analyzed with the aim to achieve more efficient use of straw for energy production. The study combines experimental research and mathematical modelling of the processes developing at co-firing of straw pellets with a low heating value biogas, which consists of 60 % methane and 40 % CO2. The complex research of the effects of straw co-firing on the gasification and combustion characteristics suggests that the increase of the biogas supply from 0 up to 1.35 l/min (additional heat input up to 0.4 kW) stabilizes the after-flame stage of the combustion of straw pellets, thus completing the burnout of volatiles, increasing the average value of the axial flow velocity, the flame temperature and the heat output from the device by about 25 – 30 %. The mathematical modelling of the straw co-firing with biogas is provided with account of variations of combustible volatiles by increasing the biogas supply into the experimental setup. A mathematical model of the combustion dynamics has been developed using MATLAB. A system of dimensionless parabolic type partial differential equations was used to describe the formation of the 2D compressible reacting swirling flow, the variations of the mass fraction of volatiles, flame species and flame temperature at thermo-chemical conversion of volatiles. Results of the numerical simulation were analyzed considering the processes developing downstream the combustor.

Modeling of the continuous casting process of steel via phase-field transition system. Fractional steps method

Here we consider the phase field transition system (a nonlinear system of parabolic type), introduced by G. Caginalp to distinguish between the phases of the material that is involved in the solidification process. On the basis of the convergence of an iterative scheme of fractional steps type, a conceptual numerical algorithm is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such approach is that the new method simplifies the numerical computations due to its decoupling feature. The finite element method (fem) in 2D is used to deduce the discrete equations and numerical results regarding the physical aspects of solidification process are reported. In order to refer the continuous casting process, the adequate boundary conditions was considered.

Approximate Controllability of Non-autonomous Evolution System with Nonlocal Conditions

In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution system of parabolic type with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. The discussions are based on Schauder’s fixed-point theorem as well as the theory of evolution family. An example is also given to illustrate the feasibility of our theoretical results.

Small-gain theorem for a class of abstract parabolic systems

We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense.

Existence and the dynamical behaviors of the positive solutions for a ratio-dependent predator–prey system with the crowing term and the weak growth

This paper deals with a ratio-dependent predator–prey system with the crowing term and the weak growth in the prey equation. Under the condition that the coefficient λ is less than a critical value λ1D(Ω0), we obtain existence of multiple positive steady state solutions of the predator–prey system and the dynamical behaviors of its positive solutions. Our results show that the predator and the prey possess not only the common coexistence, but also the very weak coexistence which both of the predator and the prey are very low. Meantime, the persistence of the positive solutions for the corresponding parabolic type system sometime depends strictly on the ratio of its initial data. Therefore, our results may be used to explain some special phenomena which under some bad environment, the predator and the prey may still coexist.

Zero relaxation limit in slow time scaling for first-order\\ quasi-linear hyperbolic systems

In this paper, we consider smooth solutions to the Cauchy problem for a multidimensional first-order quasi-linear hyperbolic system with a small parameter called relaxation time. The system is written in non-conservative form in a slow time scaling. We suppose that it is symmetrizable hyperbolic and partially dissipative. As the parameter goes to zero, it converges formally to a second-order nonlinear parabolic-type system. Under additional structural conditions on the hyperbolic system, we justify this convergence at two levels. For large initial data, we prove the local-in-time convergence of the system in a uniform time interval with respect to the parameter. When the initial data stay in a neighborhood of an equilibrium state, we prove the uniform global existence of solutions and the global-in-time convergence of the system. We also give examples of physical models to which the above results can be applied.

観測遅れを伴う放物型分布定数系に対するオブザーバ

In this note, we study the problem of designing observers for distributed parameter systems of parabolic type with sensor delay. In the work by Krstic & Smyshlyaev (2008), the case of ODE systems with sensor delay was treated and observers were derived using a backstepping method. In this note, we use a backstepping approach combined with the semigroup theory for the PDE systems mentioned above. It is shown that the sensor influence function should be chosen within some smooth functional space for observer design.

Local Dynamics of the Two-Component Singular Perturbed Systems of Parabolic Type

This paper considers the behavior of solutions from the neighborhood of an equilibrium state of nonlinear two-component parabolic problems with diffusion matrixes of one or two eigenvalues to zero. It has been shown that problems related to stability have infinite dimension. Reported here is the development of an algorithm that constructs universal families of nonlinear boundary-value problems which do not contain small parameters and whose nonlocal dynamics describes local dynamics of original boundary-value problems. In addition, an exhaustive set of universal systems for two-component parabolic equations is presented. It is concluded that a hyper multistability phenomenon is one characteristic of these systems.

Reduction, exact solutions, and control design for a certain nonlinear system of parabolic type

Reduction, exact solutions, and control design for a certain nonlinear system of parabolic type