Distributed Bilinear Systems Research Articles

Fractional output stabilization of distributed bilinear systems with a time delay

Fractional output stabilization of distributed bilinear systems with a time delay

Feedback stabilization of non-homogeneous distributed bilinear systems with varying time delay

This paper deals with the feedback stabilization for a class of non-homogeneous distributed bilinear systems, with periodically varying time delay, evolving in Hilbert state space. Sufficient conditions for polynomial and exponential stabilization are given. Explicit energy decay rates are provided in both cases. Illustrative examples are displayed.

Regional stabilization of a fractional output for a class of time-delayed distributed bilinear systems

Regional stabilization of a fractional output for a class of time-delayed distributed bilinear systems

On the feedback stabilization of distributed bilinear systems with time delay

In the present paper, we address the output stabilization for a class of infinite dimensional bilinear system with time delay on a Hilbert space. Then, we show the output weak stabilization under the weak observability condition. Furthermore, a condition like the exact observability allows exponential and strong stabilization of the output of such a system. Illustrative examples and simulations are given to validate established theoretical results.

Feedback Stabilization of Delayed Bilinear Systems

In this paper, we discuss the problem of feedback stabilization for distributed bilinear systems with discrete delay evolving on a Hilbert state space. Firstly, we prove the existence and uniqueness of the mild solution of system at hand. Secondly, we study the weak and strong stabilization. More precisely, we provide some sufficient conditions to ensure the weak and strong stabilization for the delayed bilinear system. Moreover, in the case of the strong stabilization, an explicit decay estimate is established. Applications to wave and heat equations with delayed bilinear parts are considered.

Strong and weak output stabilisation for distributed bilinear systems

This paper is concerned with the output stabilisation for a class of distributed bilinear system evolving in a spatial domain Ω. We give sufficient conditions for strong and weak output stabilisation. Also, the output stabilisation is discussed using a minimisation problem. Examples and simulations are given.

Feedback stabilisation and polynomial decay estimate for time delay bilinear systems

In this paper, we are concerned with the question of weak and strong stabilisation for distributed bilinear systems with time delay using continuous feedback controls. In the case of strong stabilisation, a polynomial decay rate of the stabilised state is estimated. Necessary and sufficient conditions for stabilisation problem are given as well. Some illustrating applications to hyperbolic and parabolic equations are considered.

Exponential and weak stabilization for distributed bilinear systems with time delay via bounded feedback control

AbstractIn this paper, we deal with the problem of exponential and weak stabilization for a class of distributed bilinear systems with time delay in a Hilbert space by using bounded feedback control. In the case of exponential stabilization, an explicit decay rate estimate of the stabilized state is given provided that a non‐standard observability inequality condition is satisfied. The compactness hypothesis of the control operator prevents the validity of the non‐standard observability inequality, thus it can be relaxed to a weaker sufficient condition to show the weak stabilization result by employing the same feedback control. Finally, numerical examples and simulations to hyperbolic and parabolic partial functional differential equations are considered to illustrate the effectiveness of the obtained theoretical results.

Stabilisation of the gradient of distributed bilinear systems

The aim of this paper is to investigate the gradient stabilisation of infinite-dimensional bilinear systems. Then under sufficient conditions, we establish exponential, strong and weak gradient stabilisation. The obtained results are illustrated by simulations.

Strong Stabilization of Distributed Bilinear Systems with Time Delay

This paper investigates global strong stabilization of time-delayed bilinear systems on a real Hilbert state space. Sufficient conditions for strong global stabilization are formulated in terms of observability like assumptions. An explicit decay rate of the stabilized state is provided. Illustrative examples are presented.

Regional gradient optimal control problem governed by a distributed bilinear systems

This paper gives an extension of previous work on gradient optimal control of distributed parabolic systems to the case of distributed bilinear systems which are a type of nonlinear systems. We introduce the notion of flux optimal control of distributed bilinear systems. The idea is trying to achieve a neighborhood of the gradient state of the considered system by minimizing a nonlinear quadratic cost. Using optimization techniques, a method showing how to reach a desired flux at a final time, only on internal subregion of the system domain will be proposed. The proposed simulation illustrates the theoretical approach by commanding the heat bilinear equation flux to a desired profile.

Feedback stabilization for unbounded bilinear systems using bounded control

Abstract In this paper, we deal with the distributed bilinear system $ \frac{d z(t)}{d t}= A z(t) + v(t)Bz(t), $ where A is the infinitesimal generator of a semigroup of contractions on a real Hilbert space H. The linear operator B is supposed bounded with respect to the graph norm of A. Then we give sufficient conditions for weak and strong stabilizations. Illustrating examples are provided.

Output stabilization of distributed bilinear systems

This paper studies regional stabilization of a distributed bilinear system evolving on a spatial domain Ω. Sufficient conditions for regional weak, strong and exponential stabilization are given. Also we discuss a regional optimal stabilization problem. The obtained results are illustrated by examples and simulations.

Polynomial decay rate estimate for bilinear parabolic systems under weak observability condition

In this paper, we shall study the stability for distributed bilinear systems on a Hilbert state space that can be decomposed in two subspaces: unstable finite-dimensional and stable infinite-dimensional with respect to the evolution generator. Then, we shall show under a weaker observability assumption that stabilizing such a system with a feedback control of the form \(p_{r}(t)=-\Vert y(t)\Vert ^{-r}\langle y(t),By(t)\rangle \) for \(r<2\), reduces stabilizing only its projection on the finite-dimension subspace which make the whole system stable. To this end, we shall propose a new family of continuous feedback controls that guarantee the uniform stabilizability with an explicit optimal decay rate estimate of the stabilized state. Two illustrating examples and simulations are provided.

Regional controllability of distributed bilinear systems

In this paper, we consider the problem of optimal regional controllability of a distributed bilinear system evolving on Ω. The question is to obtain a control with minimum energy that drives such a system from an initial state to a final state close to a desired one in finite time, only on a subregion ω of Ω. Our purpose is to prove that a regional optimal control exists and characterized in both bounded and unbounded cases. The obtained results are successfully illustrated by simulations.

Regional quadratic control problem for distributed bilinear systems with bounded controls

The aim of this paper is to study regional quadratic control problem for a distributed bilinear system evolving in a spatial domain Ω. The question is to obtain a bounded feedback control that drives such a system from an initial state to a desired one in finite time, only on a subregion ω of Ω, and minimises a quadratic functional cost. Our purpose is to prove that an optimal control exists, and characterised as solution of an optimality system. The approach is successfully illustrated by simulations.

REGIONAL QUADRATIC PROBLEM FOR DISTRIBUTED BILINEAR SYSTEMS WITH BOUNDED CONTROLS

In this paper, we establish the approximate controllability of a class of distributed bilinear systems evolving in a spatial domain Ω. A bounded feedback control is used to drive a dynamical system from an initial state to a desired one in finite time, only on a subregion ω of the system domain. Our purpose is to prove that an regional optimal control exists, and characterized as a solution to an optimality system. Numerical approach is given and successfully illustrated by simulations. AMS Subject Classification: 93B05, 93B52, 49K35

Constrained regional control problem for distributed bilinear systems

In this study, the authors consider the problem of optimal regional controllability of a distributed bilinear system evolving in a spatial domain Ω. The question is to obtain a bounded feedback control with minimum energy that drives such a system from an initial state to a desired one in finite time, only on a subregion ω of Ω. The purpose of this study is to prove that a regional optimal control exists, and characterised as a solution to an optimality system. Numerical approach is given and successfully illustrated by simulations.

Strong stabilisation and decay estimate for unbounded bilinear systems

In this article, we deal with distributed bilinear systems, where the operator of control is supposed to be unbounded in the sense that it is bounded from the state space into some extension. Then we give sufficient conditions for strong stabilisation. Also the rate of convergence of the state is estimated. An illustrating example is given.

Regional controllability for infinite-dimensional bilinear systems: approach and simulations

In this study, we consider a regional controllability problem for a class of distributed bilinear systems evolving in a spatial domain Ω. A feedback control is used to steer the system state close to a desired profile at a final time T, only on ω a subregion of the system domain which may be interior or on the boundary of Ω. Our purpose is to prove that an optimal control exists, and characterised as a solution to an optimality system. Numerical algorithm is given and successfully illustrated by simulations.