EXACT BOUNDARY CONTROLLABILITY FOR HIGHER ORDER QUASILINEAR HYPERBOLIC EQUATIONS

By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRDINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schrodinger equation posed on finite interval α≤x≤β：{ iyt ＋yzz＋｜y｜^2y = 0, y（α, t） = h1 （t）, y（β, t）=h2（t） for t〉0 （S）is considered. It is shown that by choosing appropriate control inputs （hi）, （j = 1, 2） one can always guide the system （S） from a given initial state φ∈H^S（α,β）, （s ∈ R） to a terminal state φ∈ H^s（α,β）, in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of SchrSdinger equation posed on the whole line R. The discovered smoothing properties of Schrodinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of （semi-linear） nonlinear Schrodinger equation.

Exact Boundary Controllability for a Kind of Second-Order Quasilinear Hyperbolic Systems

Based on the theory of semi-global C 1 solution and the local exact boundary controllability for first-order quasilinear hyperbolic systems,the local exact boundary controllability for a kind of second-order quasilinear hyperbolic systems is obtained by a constructive method.

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System——The Constructive Method

In this paper the authors first present the definition and some properties of weak solutions to 1-D first order linear hyperbolic systems. Then they show that the constructive method with modular structure originally given in the framework of classical solutions is still very powerful and effective in the framework of weak solutions to prove the exact boundary(null) controllability and the exact boundary observability for first order hyperbolic systems.

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem.The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method,which reduces the global exact boundary controllability problem to a local one.The proof is carried out in line with [2,15].Then a constructive method that has been developed in [13] is used to study the local problem.Especially when the region is star-complemented,it is obtained that the control function only need to be applied on a relatively open subset of the boundary.For the cubic Klein-Gordon equation,similar results of the global exact boundary controllability are proved by such an idea.

Global Exact Boundary Controllability for General First-Order Quasilinear Hyperbolic Systems

For general first-order quasilinear hyperbolic systems,based on the analysis of simple wave solutions along characteristic trajectories,the global two-sided exact boundary controllability is achieved in a relatively short controlling time.

Sufficiency of Kalman's Rank Condition for the Approximate Boundary Controllability on an Annular Domain

In this paper the author establishes the sufficiency of Kalman’s rank condition on the approximate boundary controllability at a finite time for diagonalizable systems on an annular domain in higher dimensional case.

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System——the HUM Method

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work[Lu,X.and Li,T.T.,Exact boundary con-trollability of weak solutions for a kind of first order hyperbolic system—the constructive method,Chin.Ann.Math.Ser.B,42(5),2021,643-676].In this paper,in order to study these problems from the viewpoint of duality,the authors establish a complete the-ory on the HUM method and give its applications to first order hyperbolic systems.Thus,a deeper relationship between the controllability and the observability can be revealed.Moreover,at the end of the paper,a comparison will be made between these two methods.

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions

This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.

Exact Boundary Controllability on a Tree-Like Network of Nonlinear Planar Timoshenko Beams

This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that for tree-like networks the natural initial-boundary value problem admits semi-global classical solutions in the sense of Li [Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math., vol 3,American Institute of Mathematical Sciences and Higher Education Press, 2010] existing in a neighborhood of the equilibrium solution. The authors then prove the local exact controllability of such networks near such equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.

Optimal Boundary Control Method for Domain Decomposition Algorithm

To study the domain decomposition algorithms for the equations of elliptic type, the method of optimal boundary control was used to advance a new procedure for domain decomposition algorithms and regularization method to deal with the ill posedness of the control problem. The determination of the value of the solution of the partial differential equation on the interface——the key of the domain decomposition algorithms——was transformed into a boundary control problem and the ill posedness of the control problem was overcome by regularization. The convergence of the regularizing control solution was proven and the equations which characterize the optimal control were given therefore the value of the unknown solution on the interface of the domain would be obtained by solving a series of coupling equations. Using the boundary control method the domain decomposion algorithm can be carried out.

Finite-time consensus of multi-agent systems driven by hyperbolic partial differential equations via boundary control

The leaderless and leader-following finite-time consensus problems for multiagent systems(MASs)described by first-order linear hyperbolic partial differential equations(PDEs)are studied.The Lyapunov theorem and the unique solvability result for the first-order linear hyperbolic PDE are used to obtain some sufficient conditions for ensuring the finite-time consensus of the leaderless and leader-following MASs driven by first-order linear hyperbolic PDEs.Finally,two numerical examples are provided to verify the effectiveness of the proposed methods.

Geometrical design of blade's surface and boundary control of Navier-Stokes equations

In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.

Dynamic analysis, simulation, and control of a 6-DOF IRB-120 robot manipulator using sliding mode control and boundary layer method

Because of its ease of implementation,a linear PID controller is generally used to control robotic manipulators.Linear controllers cannot effectively cope with uncertainties and variations in the parameters;therefore,nonlinear controllers with robust performance which can cope with these are recommended.The sliding mode control(SMC)is a robust state feedback control method for nonlinear systems that,in addition having a simple design,efficiently overcomes uncertainties and disturbances in the system.It also has a very fast transient response that is desirable when controlling robotic manipulators.The most critical drawback to SMC is chattering in the control input signal.To solve this problem,in this study,SMC is used with a boundary layer(SMCBL)to eliminate the chattering and improve the performance of the system.The proposed SMCBL was compared with inverse dynamic control(IDC),a conventional nonlinear control method.The kinematic and dynamic equations of the IRB-120 robot manipulator were initially extracted completely and accurately,and then the control of the robot manipulator using SMC was evaluated.For validation,the proposed control method was implemented on a 6-DOF IRB-120 robot manipulator in the presence of uncertainties.The results were simulated,tested,and compared in the MATLAB/Simulink environment.To further validate our work,the results were tested and confirmed experimentally on an actual IRB-120 robot manipulator.

Turbulent boundary layer control with a spanwise array of DBD plasma actuators

The turbulent boundary layer control on NACA 0012 airfoil with Mach number ranging from 0.3 to 0.5 by a spanwise array of dielectric barrier discharge(DBD)plasma actuators by hot-film sensor technology is investigated.Due to temperature change mainly caused through heat produced along with plasma will lead to measurement error of shear stress measured by hot-film sensor,the correction method that takes account of the change measured by another sensor is used and works well.In order to achieve the value of shear stress change,we combine computational fluid dynamics computation with experiment to calibrate the hot-film sensor.To test the stability of the hot-film sensor,seven repeated measurements of shear stress at Ma=0.3 are conducted and show that confidence interval of hot-film sensor measurement is from−0.18 to 0.18 Pa and the root mean square is 0.11 Pa giving a relative error 0.5%over all Mach numbers in this experiment.The research on the turbulent boundary layer control with DBD plasma actuators demonstrates that the control makes shear stress increase by about 6%over the three Mach numbers,which is thought to be reliable through comparing it with the relative error 0.5%,and the value is hardly affected by burst frequency and excitation voltage.

Boundary Control of Coupled Nonlinear Three Dimensional Marine Risers

This paper presents a design of boundary controllers implemented at the top end for global stabilization of a marine riser in a three dimensional space under environmental loadings. Based on the energy approach, nonlinear partial differential equations of motion, including bending-bending and longitudinal-bending couplings for the risers are derived. The couplings cause mutual effects between the three independent directions in the riser＇s motions, and make it difficult to minimize its vibrations. The Lyapunov direct method is employed to design the boundary controller. It is shown that the proposed boundary controllers can effectively reduce the riser＇s vibration. Stability analysis of the closed-loop system is performed using the Lyapunov direct method. Numerical simulations illustrate the results.

Boundary Control Problems for 2 ×2 Cooperative Hyperbolic Systems with Infinite Order Operators

In this study, boundary control problems with Neumann conditions for 2 × 2 cooperative hyperbolic systems involving infinite order operators are considered. The existence and uniqueness of the states of these systems are proved, and the formulation of the control problem for different observation functions is discussed.

An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrodinger Operator

In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.

Boundary Control for an Unstable Wave Equation in Annular with Symmetrical Initial Data

A problem of boundary stabilization of a wave equation with anti-damping term in annular is considered.This term puts some eigenvalues of the open-loop system in the right half of the complex plane.Suppose the initial and boundary conditions are rotationally symmetric,the equation in two-dimensional(2-D)annular is transformed to an equivalent one-dimensional(1-D)equation in polar coordinates.A feedback law based on the backstepping method is designed.By a successive approximation,it's proved that there exists a unique solution of the integral kernel which weights the state feedback on boundary.It's also proved that the energy function of the closed-loop system decays exponentially,implying the exponential stability of the closed-loop system.The effectiveness of the control is illustrated with numerical simulations.