Exact Controllability Problem Research Articles

Numerical method of the exact control for the elastic string problem with moving boundary

The objective of this article is to investigate, analytically and numerically, the exact control problem associated with a mathematical model that describes the vertical vibrations of elastic strings, with moving contours. To determine the approximate numerical solution, following the description of the HUM (Hilbert Uniqueness Method), the finite element method associated with the Newmark method was used for the adjunct problem and the backward problem. To calculate the optimal control, the conjugated gradient method is used. The properties of a numerical problem must be consistent with the theoretical mathematical one. However, in order to obtain the numerical control, the grid is of fundamental importance, so we used the Bi-Grid technique. In this way, we developed an algorithm to solve this problem. Numerical simulations were performed for different initial data and moving contours, with and without the Bi-Grid algorithm and the numerical results confirmed the consistency between the theoretical and numerical results.

Exact controllability for a Rayleigh beam with piezoelectric actuator

In this paper, exact controllability problem for a Rayleigh beam with piezoelectric actuator is considered. Controllability results show that the space of controllable initial data depends on the regularity of the control and the location of the actuator. Two different spaces of control, L2(0,T) and (H1(0,T))′, correspond to two different controllability properties, L2-controllability and (H1)′-controllability, respectively. The approach to prove controllability results is based on the Hilbert Uniqueness Method. Some non-controllability results are also obtained. In particular, non-controllability in short control time is studied by using Riesz basis property of exponential family in L2(0,T). Finally, minimal control time for exact controllability is obtained.

Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

We explore the simultaneous exact controllability of mean and variance of an insurance policy by utilizing the benefit St and premium Pt as control inputs to manage the policy value tV and the variance 2σt of future losses. The goal is to determine whether there exist control inputs that can steer the mean and variance from a prescribed initial state at t=0 to a prescribed final state at t=T, where the initial–terminal pair of states (0V,TV) and (2σ0,2σT) represent the mean and variance of future losses at times t=0 and t=T, respectively. The mean tV and variance 2σt are governed by Thiele’s and Hattendorff’s differential equations in continuous time and recursive equations in discrete time. Our study focuses on solving the problem of exact controllability in both continuous and discrete time. We show that our result can be used to devise control inputs St,Pt in the interval [0,T] so that the mean and variance partially track a specified curve tV=a(t) and 2σt=b(t), respectively, i.e., at a fine sampling of points in the time interval [0,T].

Hierarchical exact controllability of hyperbolic equations and Dunford-Pettis' theorem

This paper deals with the Stackelberg-Nash strategies for boundary control problems for linear and semilinear wave equations. Assuming that we can act on the system through a hierarchy of controls, to each leader, we associate a Nash equilibrium (two followers) corresponding to a bi-objective optimal control problem. Then we look for a leader that solves an exact boundary controllability problem. We mainly consider the case of a semilinear hyperbolic PDE where all the controls (leader and followers) act on small parts of the boundary. In view of the lack of regularity, the arguments usually invoked to prove controllability do not work here. Accordingly, we carry out a fixed-point approach based on weak compactness in $L^1$. To this purpose, we apply Dunford-Pettis' Theorem. We also consider the case where the followers act in small subsets of the domain.

Adaptive Fuzzy Sliding Exact Tracking Control Based on High-Order Log-Type Time-Varying BLFs for High-Order Nonlinear Systems

This article investigates the adaptive fuzzy sliding exact tracking control problem of a class of high-order nonlinear systems with full-state constraints and mismatched external disturbances. To achieve exact tracking control, the upper bound of a composite polynomial is estimated, which is composed of approximation error, mismatched disturbance, and virtual control law and its upper bound estimated information is employed in the controller design. In this way, once the tracking error deviates from the origin, the corresponding adaptive fuzzy sliding exact tracking control mechanism will be activated to force it to slide along the origin, and then exact tracking control is achieved. Meanwhile, high-order log-type time-varying barrier Lyapunov functions are introduced to ensure that the full-state constraint is not violated. A 2-D high-order nonlinear system and an underactuated nonlinear benchmark system with weak coupling are considered to illustrate the characteristics of the control mechanism in simulation.

Exact and optimal controllability for scalar conservation laws with discontinuous flux

This paper describes the reachable set and resolves an optimal control problem for the scalar conservation laws with discontinuous flux. We give a necessary and sufficient criteria for the reachable set. A new backward resolution has been described to obtain the reachable set. Regarding the optimal control problem, we first prove the existence of a minimizer and then the backward algorithm allows us to compute it. The same method also applies to compute the initial data control for an exact control problem. Our methodology for the proof relies on the explicit formula for the conservation laws with the discontinuous flux and finer properties of the characteristics curves.

On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network

In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.

A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Nodal profile control for networks of geometrically exact beams

In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of nodal profiles for the IGEB model – we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes – by using the constructive method of Zhuang, Leugering and Li (2018) [52]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network.

Global Controllability for Quasilinear Nonnegative Definite System of ODEs and SDEs

We consider exact and averaged control problem for a system of quasilinear ODEs and SDEs with a nonnegative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system and then applying the Leray–Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so-called global controllability).

Exact boundary controllability for the wave equation with moving boundary domains in a star-shaped hole

We consider an exact boundary control problem for the wave equation in a moving bounded domain which has a star-shaped hole. The boundary domain is composed by two disjoint parts, one is the boundary of the hole, which is fixed, and the other one is the external boundary which is moving. The initial data has finite energy and the control obtained is square integrable and is obtained by means of the conormal derivative. We use the method of controllability presented by Russell in [20], and assume that the control acts only in the moving part of the boundary.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/49/abstr.html

Exact Boundary Controllability and Energy Decay for a System of Wave Equations Linearly Coupled

The aim of this paper is to study an exact boundary controllability problem and local energy decay for a system of m linearly coupled wave equations . We obtain control time near of the optimal time with control of Neuman type acting on all boundary of the considered domain. We also show that the local energy decays at a rate $$t^{-n}$$ , where n is the spatial dimension of the domain where the initial data are supported.

Localization of energy and localized controllability

We will consider both the controlled Schrödinger equation and the controlled wave equation on a bounded open set Ω of ℝN during an interval of time (0, T), with T > 0. The control is distributed and acts on a nonempty open subdomain ω of Ω. On the other hand, we consider another open subdomain D of Ω and the localized energy of the solution in D. The first question we want to study is the possibility of obtaining a prescribed value of this local energy at time T by choosing the control adequately. It turns out that this question is equivalent to a problem of exact or approximate controllability in D, which we call localized controllability and which is the second question studied in this article. We obtain sharp results on these two questions and, of course, the answers will require conditions on ω and T which will be given precisely later on.

Boundary controllability of a system modelling a partially immersed obstacle

In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.

Optimization of exact controllability for fractional impulsive partial stochastic differential systems via analytic sectorial operators

Abstract In this paper, we consider the optimization problems of exact controllability for a new class of fractional impulsive partial stochastic differential systems with state-dependent delay in Hilbert spaces. By utilizing suitable fixed point approach without imposing severe compactness condition on the operators, the theory of analytic sectorial operators, stochastic analysis, and the Hausdorff measure of noncompactness, some sufficient conditions are derived for achieving the required results. Finally, an example is provided to illustrate the obtained theory.

On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.

High Order Sliding Mode Backstepping Control for a Class of Unknown Pure Feedback Nonlinear Systems

This paper aims at addressing the problem of finite time exact tracking control for a class of unknown nonlinear systems in pure feedback form while guaranteeing the closed-loop stability. The uncertain nonlinear system considered in this paper is not only in completely non-affine form but also explicitly dependent on the time and therefore, can cover a general class of nonlinear systems. By incorporating high order sliding mode controller into backstepping design procedure, a new robust control scheme with novel integrators is proposed, which is able to steer the tracking error to zero. Moreover, the closed loop stability can be proved with the help of the constructed integrators in every design step. Two simulation examples are presented to illustrate the correctness and effectiveness of the proposed control scheme.

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

<p style='text-indent:20px;'>In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the <inline-formula><tex-math id="M3">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.

Switching controls for linear stochastic differential systems

We analyze the exact controllability problem of switching controls for stochastic control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other such that, in each instant of time, there are as few active actuators as possible. We prove that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. The proof is based on building a new functional by the adjoint system whose minimizers are the switching controls.