Exact Null Controllability of a Nonlinear Thermoelastic Contact Problem

This paper considers the problem of exact null controllability of infinite dimensional linear systems in non-reflexive state spaces and reflexive control spaces. The starting point of this paper is the work of A. Vieru (Systems & Control Lett. 54 (2005) 331-337), which obtains the characterization of exact null controllability via bounded control functions in reflexive state spaces and control spaces (see Remark 2.2 of (Systems & Control Lett. 54 (2005) 331-337)). We will show here if the state space X is a general Banach space and U is reflexive, the similar result is also valid for exact null controllability via bounded control functions, which extends the results of (Systems & Control Lett. 54 (2005) 331-337).

A one-dimensional quasistatic thermoelastic contact problem with a stress-dependent boundary condition is considered. The problem models the evolution of the temperature and the displacement of a long thin elastic rod that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. The local existence of a strong solution to the problem and local dependence on the initial-boundary data are proved. In addition, the uniqueness of the solution is established. The proof rests on an abstract result dealing with perturbations of monotone operators, as well as some a priori estimates which permit an application of Schauder’s fixed point theorem.

<p style='text-indent:20px;'>This paper assesses a class of nonlocal Hilfer fractional stochastic differential equations (NHFSDEs) governed by fractional Brownian motion (fBm). New sufficient condition of exact null controllability for this stochastic setting has been investigated using fractional calculus, fixed point theory and a theory of resolvent operator. The derived result is new because it generalizes several of previously published results. However, the obtained results are proven by an illustration using stochastic partial differential equations to demonstrate the present application characteristic of null controllability. A filter example is demonstrated to make use of resolvent operator in a practical way.</p>

For a class of linear time-delay control systems satisfying the property of completability of the generalized eigenvectors, we prove that the problems of complete stabilizability and exact null controllability are equivalent.

Flutter stability is a dominant design constraint of modern turbines. Thus, flutter-tolerant designs are currently explored, where the resulting vibrations remain within acceptable bounds. In particular, friction damping has the potential to yield limit cycle oscillations (LCOs) in the presence of a flutter instability. To predict such LCOs, it is the current practice to model the aerodynamic forces in terms of aerodynamic influence coefficients for a linearized structural model with fixed oscillation frequency. This approach neglects that both the nonlinear contact interactions and the aerodynamic stiffness cause a change in the deflection shape and the frequency of the LCO. This, in turn, may have a substantial effect on the aerodynamic damping. The goal of this paper is to assess the importance of these neglected interactions. To this end, a state-of-the-art aero-elastic model of a low pressure turbine blade row is considered, undergoing nonlinear frictional contact interactions in the tip shroud interfaces. The LCOs are computed with a fully coupled harmonic balance method, which iteratively computes the Fourier coefficients of structural deformation and conservative flow variables, as well as the a priori unknown frequency. The coupled algorithm was found to provide excellent computational robustness and efficiency. Moreover, a refinement of the conventional energy method is developed and assessed, which accounts for both the nonlinear contact boundary conditions and the linearized aerodynamic influence. It is found that the conventional energy method may not predict a limit cycle oscillation at all while the novel approach presented here can.

For a linear system $$\dot x = Ax + Bu$$ with bounded operatorsA andB, exact reachability tox T or null controllability fromx 0 can be characterized by an abstract rank condition, which extends the familiar finite dimensional Kalman criterion. We provide a novel elementary proof of the more demanding sufficient part, which is based on a new device and which casts new insight even in the finite-dimensional case. Such proof provides, explicitly and constructively, an infinite class of simple, steering control functions, of preassigned smoothness, including polynomial controls. WhenA is unbounded, one cannot expect at the outset that a rank condition will play a major, in-depth description of the steering problem. Yet, our proof admits appropriate extensions to this case also. Explicit examples point out, however, the limitations of the rank conditions, as well as the new pathologies that occur.

Interior local null controllability for multi-dimensional compressible flow near a constant state

Rotational Motion Control by Feedback with Minimum L1-Norm

We consider the 2-D incompressible viscous and resistive magnetohydrodynamics (MHD) system in a rectangle, with controls on the lateral sides. The velocity satisfies Dirichlet boundary conditions, while the magnetic field follows perfectly conducting wall boundary conditions on the remaining, uncontrolled part of the boundary. We extend the small-time global exact null controllability result of Coron et al. in [Ann PDE 5(2):1-49, 2019] from Navier-Stokes equations to MHD equations, with a little help of distributed phantom forces, which can be chosen arbitrarily small in any given Sobolev spaces. Our analysis relies on Coron’s return method, the well-prepared dissipation method, long-time nonlinear Cauchy-Kovalevskaya estimates and Badra’s local exact controllability result.

Motivating the consideration of more general optimal control problems a boundary control problem for the one-dimensional heat equation with non-linear boundary condition is approximated by the Fourier method. The corresponding infinite series defining the state-function (obtained by the Fourier method) is terminated after n terms, and the set of admissible controls is replaced by a suitable approximation, too. It is shown that the sequence of “approximating” controls is strongly convergent in L 2 to a certain locally optimal control for n → ∞, provided that second order sufficient optimality conditions are fulfilled at this control and some additional assumptions on the non-linearities of the problem are satisfied. The theory is worked out for a general class of control problems governed by non-linear integral equations, which covers boundary control as a particular case.

In this paper we deal with the local exact controllability of the Navier-Stokes system with nonlinear Navier-slip boundary conditions and distributed controls supported in small sets. In a first step, we prove a Carleman inequality for the linearized Navier-Stokes system, which leads to null controllability of this system at any time T>0 . Then, fixed point arguments lead to the deduction of a local result concerning the exact controllability to the trajectories of the Navier-Stokes system.

This paper deals with a numerical study of the tribological contact stress distributions of elastomeric lip seals for oscillating shafts when the sealing interference and band width between the lip ease or contact seals and the shaft are present. Using the finite element method the contact stress and band width of scraper seals are analyzed for the sealing interference including some nonlinearities such as geometrical nonlinearity, material nonlinearity and nonlinear contact boundary condition. The FEM results showed that the contact stress concentrated on the contacting lip zone between the contacting edge of lip and the shaft for the increased interference. In double lip scraper seals, ole maximum contact stress of the dust lip, which is used to exclude foreign contaminants is six times higher than that of the primary sealing lip, which is used to contain lubricants.

Exact null controllability of semilinear integrodifferential systems in Hilbert spaces

In this paper, by using the Mountain Pass Lemma, we study the existence of nontrivial solutions for a nonlocal elliptic Kirchhoff type equation together with nonlinear boundary conditions.

Boundary element method (BEM) formulations for transient dynamic crack analysis intwo-dimensional (2D) multifield materials are reviwed in this paper. Both homogeneous and lin-ear piezoelectric as well as magnetoelectroelastic material models are considered. Special attentionis paid to properly modeling the non-linear crack-face contact and semi-permeable electric/magneticboundary conditions. Implementation of the corresponding time-domain BEM(TDBEM) is discussedin detail. The proposed TDBEM uses a Galerkin-method for the spatial discretization, whilst thecollocation method is considered for the temporal discretization. Iterative solution algorithms aredeveloped to compute the non-linear crack-face boundary conditions. Crack-tip elements that ac-count for the square-root local behavior of the crack opening displacements (CODs) at the crack-tipsare implemented. In this way, stress intensity factors (SIF), electric displacement intensity factor(EDIF) and magnetic induction intensity factor (MIIF) may be accurately evaluated from the nu-merically computed CODs at the closest nodes to the crack-tips. Numerical examples involving sta-tionary cracks in piezoelectric and magnetoelectroelastic solids under different combined (mechani-cal/electric/magnetic) impact loadings are investigated, in order to illustrate the effectiveness of theproposed approach and characterize the influence of the semi-permeable crack-face boundary condi-tions on the dynamic field intensity factors.