Critical-time boundary displacement control at two ends of an inhomogeneous rod in the case of coincidence of the wave propagation times through each of two homogeneous parts

We study a boundary displacement control at two ends of an inhomogeneous rod that has two parts of distinct densities and elasticities in the case of coinciding wave propagation times over these parts. The control acts on a time interval of critical length. We obtain a closed analytical form of the boundary displacement control bringing the rod in critical time from the initial state of rest into a given terminal state specified by given terminal displacement and terminal velocity.     

Boundary control in critical time by elastic forces applied at two ends of an inhomogeneous rod for the case in which the wave propagation time over each of the inhomogeneity parts is the same

We study boundary control in critical time by elastic forces at two ends of an inhomogeneous rod consisting of two parts of distinct densities and elasticities for the case in which the wave propagation time over each of these parts is the same. We present a closed-form expression for the boundary control by elastic forces bringing the originally quiescent stick into a given terminal state specified by a given terminal displacement and a given terminal velocity in a given critical time.     

Boundary control by an elastic force at one end of an inhomogeneous rod with the other end free for the case in which the wave travels either of the homogeneous parts in the same time

We study the boundary control by an elastic force at one end of an inhomogeneous rod that has two parts of different densities and elasticities and whose other end is free. The case in which the wave travels either of the homogeneous parts in the same time is considered. We present a closed-form analytical expression for the boundary control by an elastic force that brings the rod from the initial quiescent state to a given terminal state specified by given terminal displacement and terminal velocity.     

Optimal boundary displacement control of string vibrations with a nonlocal oddness condition of the first kind

We consider the problem of optimal boundary control by the displacement at left endpoint of a string in the case of a nonlocal oddness boundary condition of the first kind. We obtain a necessary and sufficient condition for the problem controllability under arbitrary initial and terminal conditions and construct a closed analytical form of the control itself under these conditions. In addition, we consider the problem of optimal boundary control by the displacement at one endpoint of the string for a given displacement mode at the other endpoint.     

Analytic form of optimal boundary controls of transverse vibrations of a rod consisting of several parts with different densities and elasticities but with the same impedances

We consider the problem of optimal boundary displacement control (control minimizing the boundary energy functional) of an elastic rod consisting of several parts with different densities and elasticities but with the same impedances. We obtain a closed-form expression for the optimal control bringing the rod from an arbitrarily given initial state to an arbitrarily given terminal state in a given time T.     

Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts

We optimize the boundary displacement control that is applied at one end of a rod consisting of two dissimilar parts and brings the rod vibrations from a given initial state to a given terminal state for the case in which the other end of the rod is fixed.     

Optimal boundary control of vibrations at one endpoint of a string when the second endpoint is free

The generalized solution u(x, t) of the wave equation u _tt (x, t) − u _xx (x, t) = 0 admitting the existence of finite energy at every time instant t is used to find among all W ₂ ¹ [0,T]-functions with a long time interval T the optimal boundary control for a string with a free endpoint that takes the vibration process from a given arbitrary state to a given final state. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 23–36, 2004.     

Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient. The case of control by displacement at one endpoint with the other endpoint being fixed

We consider the problem of boundary control by displacement at one boundary point x = 0 for a process described by the Klein-Gordon-Fock equation with a variable coefficient on a finite interval 0 ≤ x ≤ l with the Dirichlet condition u(l, t) = 0 at the other boundary point. For the critical time interval T = 2l, we show that there exists a unique boundary function u(0, t) = µ(t) bringing the system from an arbitrary initial state into an arbitrary terminal state.     

Optimal boundary displacement control of string vibrations with a nonlocal evenness condition of the second kind

We study the behavior of a string with the nonlocal boundary condition u _x (l, t) = u _x ($ x^\circ $ x^\circ , t). A displacement control u(0, t) = μ(t) bringing the string from an arbitrarily given initial state to an arbitrarily given terminal state is applied at the left endpoint of the string. For the initial and terminal functions, we find necessary and sufficient conditions for the controllability of the string. Under these conditions, we carry out optimization; i.e., of all admissible controls, we choose a control minimizing the boundary energy integral.     

On the vibrations described by the telegraph equation in the case of a system consisting of several parts of different densities and elasticities

We consider mixed initial-boundary value problems for longitudinal vibrations described by the telegraph equation in the case of a system consisting of several parts with different densities and elasticities but with equal impedances. We consider the cases of control by displacements at both endpoints of the rod, by elastic forces at both endpoints, and by an elastic force at one endpoint and a displacement at the other endpoint. We find closed-form expressions for the solutions of these mixed problems.     

Optimization in the class <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>2</Subscript><Superscript>2</Superscript></Stack> of the boundary control by displacements at one endpoint of a string with the other endpoint being fixed

We consider the problem on the optimal boundary control of string vibrations by a displacement at one endpoint of the string with the other endpoint being fixed. The problem is studied in the space      

Mixed problems for the string vibration equation with nonlocal conditions of the general form at the right endpoint and with an inhomogeneous condition at the left endpoint

We consider four mixed problems for the string vibration equation with zero initial conditions, with a Bitsadze–Samarskii boundary condition of the general form at the right endpoint, and with an inhomogeneous Neumann or Dirichlet condition at the left endpoint. We prove the uniqueness of a generalized solution (in the sense of Il’in) of these problems and obtain an analytic representation of these solutions. The solution of each of the problems is represented in the form of a linear combination of functions constructed from the problem data, and recursion formulas for the coefficients of this linear combination are obtained.     

Optimal boundary control of forced vibrations by the displacement at one end of the string with the other end fixed

For a string vibration process described by an inhomogeneous wave equation, we consider the problem of boundary control at one end of the string with the other end being fixed. For any time interval T > 2l, where l is the string length, we find a function u(0, t) = µ(t) bringing the vibration system from a given initial state into a given terminal state and minimizing the boundary energy integral.     

A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains

We consider an exact boundary control problem for the wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time. The optimal control that is obtained as the solution of the problem depends on the data that define the problem, in particular on the domain. Often for the numerical solution of the control problem, this given domain is replaced by a polygon. This is the motivation to study the convergence of the optimal controls for the polygon to the optimal controls for the given domain. To study the convergence, the values of the optimal controls that are defined on the boundaries of the approximating polygons are mapped in the normal directions of the polygon to control functions defined on the boundary of the original domain. This map has already been used by Bramble and King, Deckelnick, Guenther and Hinze and by Casas and Sokolowski. Using this map, we can show the strong convergence of the transformed controls as the polygons approach the given domain. An essential tool to obtain the convergence is a regularization term in the objective functions to increase the regularity of the state.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

Initial states iterative learning for three-dimensional ballistic endpoint control

In this paper, an initial states iterative learning control algorithm is proposed for control of the ballistic endpoint displacement in three-dimensional space, where the target is moving and the projectile experiences system uncertainties. The characteristics of the three-dimensional ballistic process are formulated and explored, and the learning algorithm is proposed in the spatial domain. The algorithm consists of two parts. First, the initial speed and angles are iteratively learned to make the projectile attain a fixed position. Second, the shooting time is learned to tune the arrival time of the projectile. Since the dimensions of the solution space are larger than that of the task space, three control manners, including shooting speed, shooting angle and their combination, are researched respectively. Through rigorously analyzed, it is proved that the algorithm is convergent and the multiple initial states can be adjusted simultaneously. Finally, an example of practical cannonball projection is presented to verify the effectiveness of the proposed algorithms.     

Effect of a Concentrated Normal Force on the Surface of an Orthotropic Half Space

An analytic solution is given for the stressed-deformed state of an orthotropic half space acted on by a normal concentrated force applied to its boundary. It is shown that the traditional choice of a general representation of the displacements in a gradient form, where the potential functions are in the form of double Fourier integrals with unknown densities, reduces the problem to solving two systems of linear algebraic equations – homogeneous and inhomogeneous. A solution obtained by taking a limiting transition is given for the case of an isotropic half space.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Boundary control problem for the one-dimensional Klein-Gordon-Fock equation with a variable coefficient: The case of control by displacements at two endpoints

We consider the problem of boundary control by displacements at two points x = 0 and x = l of a process described by the Klein-Gordon-Fock equation with a variable coefficient on the finite interval 0 ≤ x ≤ l. For the critical time interval T = l, we obtain a necessary and sufficient condition for the existence of unique boundary functions u(0, t) = µ(t) and u(l, t) = ν(t) bringing the system from an arbitrary initial state at t = 0 into an arbitrary terminal state at t = T.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.