Optimal boundary force control at one end of a string for a given displacement mode at the other end

We solve the problem of optimal boundary force control at one end of a string for the case of a given displacement mode at the other end. The problem is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control from infinitely many feasible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and the uniqueness of the solution is proved.     

Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.     

Optimal boundary control by displacement at one end of a string under a given elastic force at the other end

The problem of optimal boundary control by displacement at one end of a string under a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from an infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved.     

Optimization of a nonlocal boundary control of vibrations of a string with a fixed end for an arbitrary time interval that is a multiple of 2l

We consider the optimization problem for a nonlocal boundary control of vibrations of an elastic string with fixed right end for an arbitrary sufficiently large time interval that is a multiple of twice the string length. We prove the existence and uniqueness of a generalized solution of the first boundary value problem and indicate an explicit expression for the solution. The optimal control is found in closed form.     

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.     

On the nonlinear dependence of the optimal boundary control on the initial and terminal conditions

We consider the optimal boundary control of string vibrations and study the case of boundary force control at one end of the string, the other end being free. We show that if the initial and terminal data are arbitrary, then the dependence of the optimal boundary control on these data may be nonlinear. Conditions under which the dependence is linear are indicated. An example is considered in which the optimal control can be found in closed form.     

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.     

Optimal boundary control by an elastic force at one end and a displacement at the other end for an arbitrary sufficiently large time interval in the string vibration problem

We consider a boundary value problem for the wave equation with given initial conditions and with boundary conditions of the second kind at one end of the string and boundary conditions of the first kind at the other end of the string. We assume the boundary conditions to ensure that the solution of the problem (in the class of generalized functions) satisfying the initial conditions at the initial time t = 0 satisfies given terminal conditions at the terminal time t = T. We clarify the relationship between the functions µ(t) and ν(t) in the boundary conditions and the given functions specifying the initial and terminal states. We obtain closed-form analytic expressions for the functions µ(t) and ν(t) minimizing the boundary energy functional.     

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.     

Optimal boundary control of the wave equation with pointwise control constraints

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L ^∞-norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L ²-norm has a solution that we present in this paper.     

Optimization of the boundary control induced by the third boundary condition

We study the boundary control by the third boundary condition on the left end of a string, the right end being fixed. An optimality criterion based on the minimization of an integral of a linear combination of the control itself and its antiderivative raised to an arbitrary power p ≥ 1 is established. A method is developed permitting one to find a control satisfying this optimality criterion and write it out in closed form. The uniqueness of the optimal control for p > 1 is proved.     

On the optimal control of linear systems with linear performance criterion and constraints

We suggest an analytical-numerical method for solving a boundary value optimal control problem with state, integral, and control constraints. The embedding principle underlying the method is based on the general solution of a Fredholm integral equation of the first kind and its analytic representation; the method permits one to reduce the boundary value optimal control problem with constraints to an optimization problem with free right end of the trajectory.     

Solvability of the mixed problem for the wave equation with a dynamic boundary condition

We study the solvability of a mixed problem for the equation of vibrations of a bounded string with a given oblique derivative at one end in the sense of a generalized solution in the Sobolev space.     

Optimal boundary control of string vibrations by a force under elastic fixing

We study a boundary control problem based on a mixed problem with an inhomogeneous condition of the second kind at the left end of a string with elastically fixed right end. The difficulty in the solution of that problem is that the fixing condition is absent. Therefore, in addition to a constraint that is an equality of functions in the class L ₂, we need one more condition, to which V.A. Il’in refers as a condition of coordination of the initial and terminal displacements. We develop a new optimization method based on the extension of the terminal conditions to the interval [−T,T]. This permits one to minimize the integral of the squared boundary control. A control minimizing this energy integral is written out in closed form.     

Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an H^5/2-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t → + ∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions. © 2018 Wiley Periodicals, Inc.     

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.     

Inverse problem for a damped Stieltjes string from parts of spectra

We solve the following inverse problem for boundary value problems generated by the difference equations describing the motion of a Stieltjes string (a thread with beads). Given are certain parts of the spectra of two boundary value problems with two different Robin conditions at the left end and the same damping condition at the right end. From these two partial spectra, the difference of the Robin parameters, the damping constant, and the total length of the string, find the values of the point masses, and of the lengths of the intervals between them. We establish necessary and sufficient conditions for two sets of complex numbers to be the eigenvalues of two such boundary value problems and give a constructive solution of the inverse problem.     

Reduction of an elastic system to a prescribed state by use of a boundary force

A solution is constructed for the problem of optimal control of the motion of a distributed elastic system, using a lumped boundary force. The system's state is described by a constant-coefficient hyperbolic equation. A general case of arbitrary initial and final distributions is examined. Questions of control by lumped and distributed forces are discussed.     

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 control of an elastic string with viscous damping

We study bilinear optimal control of a wave equation with one spatial dimension. The problem describes oscillations of an elastic string with viscous damping, and the damping coefficient is taken as the control. The objective functional involves driving the state solution close to a desired profile and incurring a cost on the control. The optimal control is characrerized in terms of an optimality system.