Optimal boundary feedback stabilization of a string with moving boundary

Email: gugat{at}am.uni-erlangen.de Received on April 30, 2006; We consider a finite string that is fixed at one end and subjectto a feedback control at the other end which is allowed to move.We show that the behaviour is similar to the situation whereboth ends are fixed: As long as the movement is not too fast,the energy decays exponentially and for a certain parameterin the feedback law it vanishes in finite time. We considermovements of the boundary that are continuously differentiablewith a derivative whose absolute value is smaller than the wavespeed. We solve a problem of worst-case optimal feedback control,where the parameter in the feedback law is chosen such thatthe worst-case L^p-norm of the space derivative at the fixedend of the string is minimized (p [1, )). We consider the worstcase both with respect to the initial conditions and with respectto the boundary movement. It turns out that the parameter forwhich the energy vanishes in finite time is optimal in thissense for all p.     

Optimal control problem for viscous Cahn-Hilliard equation

This paper is concerned with the viscous Cahn-Hilliard equation, which arises in the dynamics of viscous first order phase transitions in cooling binary solutions. The optimal control under boundary condition is given and the existence of optimal solution to the equation is proved.     

Optimal control asymptotics of a magnetohydrodynamic flow

An optimal multiplicative control problem is considered for a one-dimensional magnetohydrodynamic flow between parallel planes (Hartman flow). The external magnetic field is used as a control function. An optimality system is derived, and the asymptotics of an optimal control with respect to a regularization parameter and the Reynolds number are constructed.     

Optimal control of the viscous weakly dispersive Degasperis-Procesi equation

In this paper, we study the optimal control problem for the viscous weakly dispersive Degasperis-Procesi equation. We deduce the existence and uniqueness of a weak solution to this equation in a short interval by using the Galerkin method. Then, according to optimal control theories and distributed parameter system control theories, the optimal control of the viscous weakly dispersive Degasperis-Procesi equation under boundary conditions is given and the existence of an optimal solution to the viscous weakly dispersive Degasperis-Procesi equation is proved.     

具有劳动力增长的资产发展方程的最优控制问题

经济系统是一个复杂巨系统,具有复杂的层次结构.近年来,系统科学理论的发展为研究经济系统提供了新的思路和方法,已有了很大进展.劳动力是资产发展系统中的一个重要参数,对具有劳动力增长的非线性资产发展方程中劳动力的最优控制问题进行了研究.利用Banach空间理论,对极小化序列中的弱收敛序列,构造一强收敛极小化序列,得到了其最优解的存在性和唯一性,结果推广和改进了最近文献的一些主要结果.这个问题的研究对于促进我国经济高速、稳定持续增长具有重要的理论意义和现实指导价值.     

Optimal control of the multidimensional diffusion equation

The existence and numerical estimation of a boundary control for then-dimensional linear diffusion equation is considered. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures. The existence of an optimal measure corresponding to the above problem is shown, and the optimal measure is approximated by a finite convex combination of atomic measures. This construction gives rise to a finite-dimensional linear programming problem, whose solution can be used to construct the combination of atomic measures, and thus a piecewise-constant control function which approximates the action of the optimal measure, so that the final state corresponding to the above control function is close to the desired final state, and the value it assigns to the performance criterion is close to the corresponding infimum. A numerical procedure is developed for the estimation of these controls, entailing the solution of large, finite-dimensional linear programming problems. This procedure is illustrated by several examples.     

Exact controllability of the heat equation with bilinear control

This paper deals with exact controllability of bilinear heat equation. Namely, given the initial state, we would like to provide a class of target states that can be achieved through the heat equation at a finite time by applying multiplicative controls. For this end, an explicit control strategy is constructed. Simulations are provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces

Email: vio{at}utgjiu.ro Received on September 12, 2007; Accepted on December 26, 2008 In this article, we discuss a quadratic control problem forlinear discrete-time systems with Markov perturbations in Hilbertspaces, which is linked to a discrete-time Riccati equationdefined on certain infinite-dimensional ordered Banach space.We prove that under stabilizability and stochastic uniform observabilityconditions, the Riccati equation has a unique, uniformly positive,bounded on N and stabilizing solution. Based on this result,we solve the proposed optimal control problem. An example illustratesthe theory.     

Vibrations of elastic string with nonhomogeneous material

In this paper we analyze from the mathematical point of view a model for small vertical vibrations of an elastic string with fixed ends and the density of the material being not constant. We employ techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of Faedo-Galerkin method. We obtain strong global solutions with restrictions on the initial data u₀ and u₁, uniqueness of solutions and a rate decay estimate for the energy.     

Controllability of the wave equation with moving point control

This paper deals with approximate and exact controllability of the wave equation in finite time with interior point control acting along a curve specified in advance in the system's spatial domain. The structure of the control input is dual to the structure of the observations which describe the measurements of velocity and gradient of the solution of the dual system, obtained from the moving point sensor. A relevant formalization of such a control problem is discussed, based on transposition. For any given timeinterval [0,T] the existence of the curves providing approximate controllability inH _D ^–[n/2]–1 ()×H _D ^–[n/2]–1 () (wheren stands for the space dimension) is established with controls fromL ²(0,T; R ⁿ +1). The same curves ensure exact controllability inL ²() × H^–1() if controls are allowed to be selected in [L (0,T; R ⁿ+1)]. Required curves can be constructed to be continuous on [0,T).This work was supported in part by NSF Grant ECS 89-13773 and NASA Grant NAG-1-1081.     

Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems

We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is described by the dynamics of a pendulum that is fixed to a trolley moving along the crane beam. The resulting problem serves as a case study for optimal control of fully coupled partial and ordinary differential equations (ODEs). This particular type of coupled systems arises from many applications involving mechanical multi-body systems. We motivate the coupled ODE-PDE model, show its analytical well-posedness locally in time and examine the corresponding optimal control problem numerically by means of a projected gradient method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) update.     

On an exact control problem for a semilinear wave equation with an unknown source

The exact controllability of a semilinear wave equation, with Dirichlet boundary control on a part of the boundary and an unknown source, is shown. The nonlinear term has at most a linear growth, the initial and target spaces are L²(Ω)×H⁻¹(Ω).     

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.     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non‐uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal control for a sixth order nonlinear parabolic equation

In this paper, we consider the optimal control problem for a sixth order nonlinear parabolic equation, which arising in oil‐water‐surfactant mixtures. Based on Lions' theory, we prove the existence of optimal solution. The optimality system is also established. Copyright © 2013 John Wiley & Sons, Ltd.     

Uniform stability of damped nonlinear vibrations of an elastic string

Here we are concerned about uniform stability of damped nonlinear transverse vibrations of an elastic string fixed at its two ends. The vibrations governed by nonlinear integro-differential equation of Kirchoff type, is shown to possess energy uniformly bounded by exponentially decaying function of time. The result is achieved by considering an energy-like Lyapunov functional for the system.     

Optimal control of non-isothermal viscous fluid flow

For flow inside a four-to-one contraction domain, we minimize the vortex that occurs in the corner region by controlling the heat flux along the corner boundary. The problem of matching a desired temperature along the outflow boundary is also considered. The energy equation is coupled with the mass, momentum, and constitutive equations through the assumption that viscosity depends on temperature. The latter three equations are a non-isothermal version of the three-field Stokes–Oldroyd model, formulated to have the same dependent variable set as the equations governing viscoelastic flow. The state and adjoint equations are solved using the finite element method. Previous efforts in optimal control of fluid flows assume a temperature-dependent Newtonian viscosity when describing the model equations, but make the simplifying assumption of a constant Newtonian viscosity when carrying out computations. This assumption is not made in the current work.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.