Asymptotic Null Controllability Research Articles

Application of Non-Linear Evolution Stochastic Equations with Asymptotic Null Controllability Analysis

This paper investigated system of stochastic differential equations with prominence on disparities of drift parameters. These problems were solved analytical by adopting the Ito’s method of solution and three different investment solutions were obtained consequently. The necessary conditions were achieved which govern various drift parameters in assessing financial markets. Therefore, the impressions on each solution of investors in financial markets were analyzed graphically. Secondly, stock price data of Transco, LTD were analyzed which covariance matrix were considered and analysis were logically extended to stochastic vector differential equation where control measures were incorporated that would help in predicting different stock price processes, and the result obtained by exploring the properties of the fundamental matrix solution where asymptotic null controllability results were obtained by the singularity of the controllability matrix a function of the drift. Finally, the effects of the significant parameters of stochastic variables were successfully discussed.

Partial asymptotic null controllability for semilinear evolution equations in Hilbert spaces

This work deals with the problem of partial asymptotic null controllability for constrained systems governed by semi-linear evolution equations, where the linear part generates a strongly continuous compact semigroup on Hilbert spaces. This consists of driven only a desired part of the system's state to the origin, taking into account mixed state-input constraints. We give sufficient conditions for the existence of appropriate control laws to ensure partial asymptotic null controllability. The explicit expression of such controls is given as strongly-weakly continuous selections of a set-valued map which is defined through a certain practical tangential condition. Application to semilinear parabolic partial differential equations is treated in order to illustrate the results established.

Partial asymptotic null-controllability with mixed state-input constraints

This paper is devoted to the problem of partial asymptotic null-controllability of control systems governed by ordinary differential equations, subjected to possibly mixed state-input constraints. Using Lyapunov functions within the framework of viability theory, feedback controls are designed in such a way a part of system's state can be driven to the origin asymptotically, taking into account the mixed constraints. By using Michael selection theorem, the existence of such controls is proved, in the case of convex constraints, and their expressions are given as continuous selections of an appropriate constructed multifunction. Finally, two examples are processed numerically in order to illustrate the theoretical results.

Constrained asymptotic null-controllability for semi-linear infinite dimensional systems

This paper investigates asymptotic controllability of systems governed by semi-linear partial differential equations, under mixed input-state constraints. A unified approach based on viability theory and set-valued analysis, is provided, requiring that the linear part of the system generates a strongly continuous compact semigroup. In the case of convex constraints, it is shown that Michael selection theorem can be used to get existence of the needed feedback control laws. Examples are numerically treated in order to illustrate the results established.

State‐input constrained asymptotic null‐controllability by a set‐valued approach

This study presents a unified approach to investigate asymptotic null-controllability under possibly mixed constraints on state and control, for standard ordinary differential equation control systems. Using tools from set-valued analysis and viability theory, initial data which can be steered to the origin at infinity are provided through a new type of Lyapunov functions. The corresponding controls are given via selections of adequately designed multifunctions, which are examined in both cases of convex constraints and the class of affine-control systems. Finally, numerical examples from classical mechanics are given to verify the theoretical results.

Contrôlabilité asymptotique et synchronisation asymptotique dʼun système couplé dʼéquations des ondes avec des contrôles frontières de Dirichlet

In this Note, the asymptotic null controllability and various kinds of asymptotic synchronization, considered as some kinds of weakened controllability and synchronization, are introduced and studied for a coupled system of wave equations with Dirichlet boundary controls. Equivalent properties of weak observability are established.

Contrôlabilité asymptotique de systèmes hyperboliques linéaires

In this Note we introduce the asymptotic controllability and the asymptotic null controllability for 1- D linear hyperbolic systems under the lack of boundary controls. We claim that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4 × 4 hyperbolic system with only one boundary control is shown to be asymptotically controllable but not exactly controllable.

Control Lyapunov Functions and Zubov's Method

For finite-dimensional nonlinear control systems we study the relation between asymptotic null-controllability and control Lyapunov functions. It is shown that control Lyapunov functions (CLFs) may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubov's equation. The solution is also given as the value function of an optimal control problem from which several regularity results may be obtained.

Construction of lyapunov functions on the domain of asymptotic nullcontrollability: Numerics

We present a numerical method for the computation of control Lyapunov functions on the domain of nullcontrollability of a nonlinear system. We apply an adaptive semi-Lagrangian discretization technique to a generalized version of the Zubov equation whose solutions provide such Lyapunov functions. In particular, we address regularization issues which need to be resolved before the scheme is applicable and discuss an adaptive space discretization technique.

Construction of lyapunov functions on the domain of asymptotic nullcontrollability: Theory

In this paper we study the domain of asymptotic nullcontrollability of control systems. It is shown that this domain may be characterized by optimal control problems whose value function is the unique viscosity solution of a suitable first-order PDE. This value function turns out to be a control Lyapunov function. The approach generalizes Zubov's method.

Input-to-state stability of exponentially stabilized semilinear control systems with inhomogeneous perturbations

In this paper we investigate the robustness of state feedback stabilized semilinear systems subject to inhomogeneous perturbations in terms of input-to-state stability. We consider a general class of exponentially stabilizing feedback controls which covers sampled discrete feedbacks and discontinuous mappings as well as classical feedbacks and derive a necessary and sufficient condition for the corresponding closed-loop systems to be input-to-state stable with exponential decay and linear dependence on the perturbation. This condition is easy to check and admits a precise estimate for the constants involved in the input-to-state stability formulation. Applying this result to an optimal control based discrete feedback yields an equivalence between (open-loop) asymptotic null controllability and robust input-to-state (state feedback) stabilizability.

Feedback stabilization of discrete-time homogeneous semi-linear systems

For discrete-time semi-linear systems satisfying an accessibility condition asymptotic null controllability is equivalent to exponential feedback stabilizability using a piecewise constant feedback. A constructive procedure that yields such a feedback is presented.

Discrete feedback stabilization of semilinear control systems

For continuous time semilinear control systems with constrained control values stabilizing discrete feedback controls are discussed. It is shown that under an accessibility condition exponential discrete feedback stabilizability is equivalent to open loop exponential asymptotic null controllability. A numerical algorithm for the computation of discrete feedback controls is presented and a numerical example is discussed.

Asymptotic null controllability of bilinear systems

Asymptotic null controllability of bilinear systems

Asymptotic null controllability of nonlinear perturbed systems

In this paper, sufficient conditions are obtained for the asymptotic null controllability of the system $$\dot x(t) = g(t,x(t)) + B(t,x(t))u(t) + f(t,x(t),u(t)).$$