Control Of Parabolic Systems Research Articles

Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix

The distributed null controllability for coupled parabolic systems with non-diagonalizable diffusion matrices with a reduced number of controls has been studied in the case of constant matrices. On the other hand, boundary controllability issues and distributed controllability with non-constant coefficients for this kind of systems is not completely understood. In this paper, we analyze the boundary controllability properties of a class of coupled parabolic systems with non-diagonalizable diffusion matrices in the constant case and the distributed controllability of a $ 2\times 2 $ non-diagonalizable parabolic system with space-dependent coefficients. For the boundary controllability problem, our strategy relies on the moment method. For the distributed controllability problem, our findings provide positive and negative control results by using the Fattorini-Hautus test and a fictitious control strategy.

Results on Boundary Control for Parabolic Systems Using Backstepping Method

This paper focuses on the stabilization problem for the linear parabolic system using the backstepping method. The exponentially stability results for considered parabolic system are derived in two cases with Dirichlet and Neumann local terms. Also, the boundary conditions for the problem is assumed to be mixed or Robin-type boundary conditions. The main aim is to achieve the stability of the considered system using the backstepping method with help of Volterra integral transformation. The explicit solutions of kernel functions in integral transformation is obtained by using Laplace transform and designed a boundary control law to the closed-loop system. Finally, the effectiveness and applicability of the derived results are validated through a single-species pattern generation model.

Time optimal control to a partial stochastic differential system with pseudo almost periodic coefficients

In this paper, we study the time optimal control of a class of partial stochastic impulsive functional differential systems with pseudo almost periodic coefficients in the α-norm functional spaces. Firstly, the existence of p-mean piecewise pseudo almost periodic mild solutions for the impulsive stochastic control systems is investigated by utilising stochastic analysis, analytic semigroup, the fixed-point techniques with fractional power arguments. Secondly, the existence of time optimal pairs of a system governed by partial stochastic impulsive functional differential equations is also obtained. Finally, two examples are provided to illustrate the time optimal control problems of parabolic control system with pseudo almost periodic coefficients and impulses.

Time-optimality by distance-optimality for parabolic control systems

The equivalence of time-optimal and distance-optimal control problems is shown for a class of parabolic control systems. Based on this equivalence, an approach for the efficient algorithmic solution of time-optimal control problems is investigated. Numerical examples are provided to illustrate that the approach works well in practice.

Stabilizability of Time-Periodic Evolution Equations by Finite Dimensional Controls

We study the stabilizability of time-periodic parabolic control systems with unbounded control operators. We give necessary and sufficient conditions for stabilizability which extend to the case of unbounded control operators, and which sharpen, conditions existing in the literature in the case of bounded control operators. We further show how to construct families of finite dimensional controls which preserves the stabilizability of the original system.

Local feedback stabilization of time-periodic evolution equations by finite dimensional controls

We study the feedback stabilization around periodic solutions of parabolic control systems with unbounded control operators, by controls of finite dimension. We prove that the stabilization of the infinite dimensional system relies on the stabilization of a finite dimensional control system obtained by projection and next transformedviaits Floquet-Lyapunov representation. We emphasize that this approach allows us to define feedback control laws by solving Riccati equations of finite dimension. This approach, which has been developed in the recent years for the boundary stabilization of autonomous parabolic systems, seems to be totally new for the stabilization of periodic systems of infinite dimension. We apply results obtained for the linearized system to prove a local stabilization result, around periodic solutions, of the Navier-Stokes equations, by finite dimensional Dirichlet boundary controls.

Optimal control for boundary-degenerate parabolic equations with logistic growth

This paper deals with the optimal control problem of a parabolic control system with the logistic growth and the boundary degeneracy.We firstly establish the well-posedness of weak solutions. Then the existence and the stabilityof the optimal control that minimizes the cost functional are proved. Moreover, the optimal controlis characterized in terms of the unique solution of the optimality system, which consists of thestate equation coupled with the adjoint equation.

Optimal controls for impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients

In this paper, we study optimal controls of a class of impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients in Hilbert spaces. Firstly, using interpolation theory, the stochastic analysis techniques and a fixed-point theorem, the existence of p-mean piecewise weighted pseudo almost periodic in distribution mild solutions for the impulsive stochastic control system is presented. Secondly, the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, application to optimal control problems of impulsive parabolic control system is considered.

On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

Periodic solutions for a class of evolution inclusions

We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.

The Controllability of Parabolic Systems with Delay and Distributed Parameters on the Graph

The optimal boundary control problem for a differential system with delay and distributed parameters on the graph is considered. As a state-space systems and the space of boundary control and observation, the classes of integrable functions is used. The conditions of the unique solvability of the problem of optimal control are obtained. And finally, the existence conditions of a unique control action and the controllability system are obtained.

Оптимальное управление параболической системой с распределенными параметрами по графе в пространстве решений с производной времени

Оптимальное управление параболической системой с распределенными параметрами по графе в пространстве решений с производной времени

Algebraic Multiplicities Arising from Static Feedback Control Systems of Parabolic Type

In stabilization studies of linear parabolic control systems, a successful approach is a scheme employing dynamic compensators in the feedback loop. An essential reason is the fact that both sensors and actuators cannot be designed freely, especially in the case of boundary observation/boundary feedback. Most fundamental in this scheme is a simple stabilization result under the static feedback control scheme. In this scheme, little attention has been paid to how to assign new eigenvalues of the feedback system. In this article, we show a new feature of pole assignment that shows some choices of new eigenvalues cause a deterioration of the stability property. An algebraic growth rate is added to the feedback system in such a choice.

Alternative algebraic approach to stabilization for linear parabolic boundary control systems

In this paper, we develop an alternative algebraic approach to feedback stabilization problems of linear parabolic boundary control systems. The control system is general in the sense that no Riesz basis associated with the elliptic operator is expected. The control system contains a finite dimensional dynamic compensator, the dimension of which has been so far determined only by the actuators on the boundary. One of the purposes of the paper is to show a new alternative scheme that the dimension of the compensator could be determined by the sensors on the boundary: it leads to a realization of stabilizing compensators of lower dimension by comparison. This is achieved by introducing a new operator equation connecting two states of the controlled plant and the compensator. The other purpose is a generalization of the condition posed on the sensors. When the elliptic operator admits generalized eigenfunctions, the sharpest criterion is proposed on the minimum choice of the necessary number of the sensors. In fact, only one sensor is enough for the stabilization in the best case (the algebraic multiplicities of the corresponding eigenvalues may, of course, be greater than 1).

On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems

Unlike regular stabilizations, we construct in the paper a specific feedback control system such that u(t) decays exponentially with the designated decay rate, and that some non-trivial linear functionals of u decay exactly faster than . The system contains a dynamic compensator with another state v in the feedback loop, and consists of two states u and v. This problem entirely differs from the one with static feedback scheme in which the system consists only of a single state u. To show the essential difference, some specific property of the spectral subspaces associated with our control system is studied.

Balancing-Related Model Reduction for Parabolic Control Systems

Balanced truncation (BT) is a well established model reduction technique for linear ordinary differential equations. If the original dynamics are described by an instationary PDE, then BT is usually applied to the large-scale linear system resulting from a spatial semi-discretization using finite elements/volumes/differences. We will discuss this approach as well as a variant of BT based on balancing the solution of the linear-quadratic Gaussian (LQG) algebraic Riccati equations, called LQG BT, allowing the reduction of unstable systems and yielding a stabilizing feedback controller as a by-product. The error between reduced-order model and original system is split into discretization and model reduction components. We discuss the resulting error bounds and ways how to exploit this in order to adaptively choose the reduced system order. Numerical examples support our findings.

Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations

In this paper, we study optimal relaxed controls and relaxation of nonlinear fractional impulsive evolution equations. Firstly, existence of piecewise continuous mild solutions for the original fractional impulsive control system is presented. Secondly, fractional impulsive relaxed control system is constructed by using a regular countably additive measure and making the original control system convexified. Thirdly, optimal relaxed controls and relaxation theorems are obtained. Finally, application to initial-boundary value problem of fractional impulsive parabolic control system is considered.

Equivalence of Two Stabilization Schemes for a Class of Linear Parabolic Boundary Control Systems

Equivalence of Two Stabilization Schemes for a Class of Linear Parabolic Boundary Control Systems

A Note on the Minimum Number of the Actuators for Stabilization in Linear Parabolic Boundary Control Systems

In the stabilization problem for linear boundary control systems of parabolic type, we have just recently obtained a criterion on the smallest number of the sensors. We show in this note that a sim...

Optimal control and feedback design of state-constrained parabolic systems in uncertainty conditions

This article concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in the Dirichlet boundary conditions. The underlying parabolic control system is functioning under hard/pointwise constraints on control and state variables. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We develop a constructive approach to the minimax control design of constrained parabolic systems that is based on certain characteristic features of the parabolic dynamics, including the transient monotonicity with respect to both controls and perturbations and the turnpike asymptotic behaviour on the infinite horizon. In this way, solving a number of associated open-loop control and optimization problems, we justify an easily implementable three-positional suboptimal structure of the feedback boundary regulator and compute its optimal parameters, thus ensuring the required state performance and robust stability of the closed-loop, highly nonlinear parabolic control system on the infinite horizon.