arXiv subjects

Ahmet Ozkan Ozer

Publications and source records attributed to Ahmet Ozkan Ozer.

16 recordsLinked to original sources

arXiv2025

Two Strings with a Dynamic Interior Mass: A Feedback Control Design with Guaranteed Exponential Decay

This paper investigates the exponential stabilization of a coupled two-string system joined by a dynamic interior mass. The combined effect of three feedback mechanisms, boundary damping from tip velocity, higher-order nodal damping from angular velocity, and lower-order nodal damping from mass velocity, is analyzed using a Lyapunov framework. Exponential stability is established unconditionally, without constraints on wave speeds or mass location, improving upon earlier results that lower-order nodal damping, as in {Hansen-Zuazua'95}, or boundary damping alone, as in {Lee-You'89}, does not ensure exponential decay without additional structural conditions. Moreover, the lower-order feedback can be removed without loss of exponential decay when combined with the other two mechanisms, via a compact perturbation argument. These results apply to hybrid systems with interior or tip mass interfaces, including overhead cranes, deep-sea cables, and fluid structure interaction. Theoretical findings are validated through numerical simulations.

math.AP↗

arXiv2025

Robust Model Reductions for the Boundary Feedback Stabilization of Magnetizable Piezoelectric Beams

Magnetizable piezoelectric beams exhibit strong couplings between mechanical, electric, and magnetic fields, significantly affecting their high-frequency vibrational behavior. Ensuring exponential stability under boundary feedback controllers is challenging due to the uneven distribution of high-frequency eigenvalues in standard Finite Difference models. While numerical filtering can mitigate instability as the discretization parameter tends to zero, its reliance on explicit spectral computations is computationally demanding. This work introduces two novel model reduction techniques for stabilizing magnetizable piezoelectric beams. First, a Finite Element discretization using linear splines is developed, improving numerical stability over standard Finite Differences. However, this method still requires numerical filtering to eliminate spurious high-frequency modes, necessitating full spectral decomposition. Numerical investigations further reveal a direct dependence of the optimal filtering threshold on feedback amplifiers. To overcome these limitations, an alternative order-reduction Finite Difference scheme is proposed, eliminating the need for numerical filtering. Using a Lyapunov-based framework, we establish exponential stability with decay rates independent of the discretization parameter. The reduced model also exhibits exponential error decay and uniform energy convergence to the original system. Numerical simulations validate the effectiveness of the proposed methods, and we construct an algorithm for separating eigenpairs for the proper application of the numerical filtering. Comparative spectral analyses and energy decay results confirm the superior stability and efficiency of the proposed approach, providing a robust framework for model reduction in coupled partial differential equation systems.

math.NA↗

arXiv2024

Exponential Stability and Design of Sensor Feedback Amplifiers for Fast Stabilization of Magnetizable Piezoelectric Beam Equations

The dynamic partial differential equation (PDE) model governing longitudinal oscillations in magnetizable piezoelectric beams exhibits exponentially stable solutions when subjected to two boundary state feedback controllers. An analytically established exponential decay rate by the Lyapunov approach ensures stabilization of the system to equilibrium, though the actual decay rate could potentially be improved. The decay rate of the closed-loop system is highly sensitive to the choice of material parameters and the design of the state feedback amplifiers. This paper focuses on investigating the design of state feedback amplifiers to achieve a maximal exponential decay rate, which is essential for effectively suppressing oscillations in these beams. Through this design process, we explicitly determine the safe intervals of feedback amplifiers that ensure the theoretically found maximal decay rate, with the potential for even better rates. Our numerical results reaffirm the robustness of the decay rate within the chosen range of feedback amplifiers, while deviations from this range significantly impact the decay rate. To underscore the validity of our results, we present various numerical experiments.

math.OC↗

arXiv2024

Boundary Output Feedback Stabilization for a Novel Magnetizable Piezoelectric Beam Model

A magnetizable piezoelectric beam model, free at both ends, is considered. Piezoelectric materials have a strong interaction of electromagnetic and acoustic waves, whose wave propagation speeds differ substantially. The corresponding strongly-coupled PDE model describes the longitudinal vibrations and the total charge accumulation at the electrodes of the beam. It is known that the PDE model with appropriately chosen collocated state feedback controllers is known to have exponentially stable solutions. However, the collocated controller design is not always feasible since the performance of controllers may not be good enough, and moreover, a small increment of feedback controller gains can easily make the closed-loop system unstable. Therefore, a non-collocated controller and observer design is considered for the first time for this model. In particular, two state feedback controllers are designed at the right end to recover the states so that the boundary output feedback controllers can be designed as a replacement of the states with the estimate from the observers on the left end. By a carefully-constructed Lyapunov function, it is proved that the both the observer and the observer error dynamics have uniformly exponential stable solutions. This framework offers a substantial foundation for the model reductions by Finite Differences.

math.AP↗

arXiv2024

The Exponential Stabilization of a Heat and Piezoelectric Beam Interaction with Static or Hybrid Feedback Controllers

This study investigates a strongly-coupled system of partial differential equations (PDE) governing heat transfer in a copper rod, longitudinal vibrations, and total charge accumulation at electrodes within a magnetizable piezoelectric beam. Conducted within the transmission line framework, the analysis reveals profound interactions between traveling electromagnetic and mechanical waves in magnetizable piezoelectric beams, despite disparities in their velocities. Findings suggest that in the open-loop scenario, the interaction of heat and beam dynamics lacks exponential stability solely considering thermal effects. To confront this challenge, two types of boundary-type state feedback controllers are proposed: (i) employing static feedback controllers entirely and (ii) adopting a hybrid approach wherein the electrical controller dynamically enhances system dynamics. In both cases, solutions of the PDE systems demonstrate exponential stability through meticulously formulated Lyapunov functions with diverse multipliers. The proposed proof technique establishes a robust foundation for demonstrating the exponential stability of Finite-Difference-based model reductions as the discretization parameter approaches zero.

math.OC↗

arXiv2023

A Novel Sensor Design for a Cantilevered Mead-Marcus-type Sandwich Beam Model by the Order-reduction Technique

A novel space-discretized Finite Differences-based model reduction, introduced in (Liu,Guo,2020) is extended to the partial differential equations (PDE) model of a multi-layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE can not be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the ``order-reduced'' Finite-Differences technique. This method does not require any type of filtering though the exact observability as $h\to 0$ is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.

math.OC↗

arXiv2023

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

The one-dimensional PDE model of the wave equation with a state feedback controller at its boundary, which describes wave dynamics of a wide-range of controlled mechanical systems, has exponentially stable solutions. However, it is known that the reduced models of the wave equation by the standard Finite Differences and Finite Elements suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, to which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough eigenvalue estimation.The maximal decay rate is explicitly provided as a function of the feedback gain and filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results.

math.NA↗

arXiv2018

Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results

A cantilevered piezoelectric smart composite beam, consisting of perfectly bonded elastic, viscoelastic and piezoelectric layers, is considered. The piezoelectric layer is actuated by a voltage source. Both fully dynamic and electrostatic approaches, based on Maxwell's equations, are used to model the piezoelectric layer. We obtain (i) fully-dynamic and electrostatic Rao-Nakra type models by assuming that the viscoelastic layer has a negligible weight and stiffness, (ii) fully-dynamic and electrostatic Mead-Marcus type models by neglecting the in-plane and rotational inertia terms. Each model is a perturbation of the corresponding classical smart composite beam model. These models are written in the state-space form, the existence and uniqueness of solutions are obtained in appropriate Hilbert spaces. Next, the stabilization problem for each closed-loop system, with a thorough analysis, is investigated for the natural $B^*-$type state feedback controllers. The fully dynamic Rao-Nakra model with four state feedback controllers is shown to be not asymptotically stable for certain choices of material parameters whereas the electrostatic model is exponentially stable with only three state feedback controllers (by the spectral multipliers method). Similarly, the fully dynamic Mead-Marcus model lacks of asymptotic stability for certain solutions whereas the electrostatic model is exponentially stable by only one state feedback controller.stable by one feedback controller applied to the bending moment only.

math.AP↗

arXiv2018

Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams

Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piezoelectric beam with and without the electromagnetic interactions (fully dynamic and electrostatic). This modeling strategy is extended for the three-layer piezoelectric smart composites by adopting the two widely-accepted sandwich beam theories. For both single-layer and three-layer models, the resulting infinite dimensional equations of motion can be formulated in the state-space form. It is observed that the fully dynamic nonlinear models are unbounded boundary control systems (same in linear theory) ${\bf \dot y}(t)=(\mathcal A +\mathcal N) {\bf y} (t) + \mathcal B u(t)$, the electrostatic nonlinear models are unbounded bilinear control systems ${\bf \dot y}(t)=(\mathcal A +\mathcal N){\bf y} (t) + (\mathcal B_1+ \mathcal B_2 {\bf y}) u(t)$ in sharp contrast to the linear theory. Finally, we propose $\mathcal B^*-$type feedback controllers to stabilize the single piezoelectric beam models. The filtered semi-discrete Finite Difference approximations is adopted to illustrate the findings.

math.OC↗

arXiv2018

Exponential stabilization of a smart piezoelectric composite beam with only one boundary controller

Layered smart composite beams involving a piezoelectric layer are traditionally actuated by a voltage source by the extension mechanism. In this paper, we consider only the bending and shear of a cantilevered piezoelectric smart composite beam modeled by the Mead-Marcus sandwich beam assumptions. Uniform exponential stabilitization with only one boundary state feedback controller, simultaneously controlling both bending moment and shear, is proved by using a spectral multiplier approach. The state feedback controller slightly differs from the classical counterparts by a non-trivial compact and nonnegative integral operator. This is due to the strong coupling of the charge equation with the stretching and bending equations. For simulations, the so-called filtered semi-discrete finite difference scheme is adopted.

math.AP↗

arXiv2017

Modeling and controlling an active constrained layer (ACL) beam actuated by two voltage sources with/without magnetic effects

A fully dynamic three-layer active constrained layer (ACL) beam is modeled for cantilevered boundary conditions by using a thorough variational approach. The Rao-Nakra thin compliant layer assumptions are adopted to model the sandwich structure, and all magnetic effects for the piezoelectric layers are retained. The piezoelectric layers are activated by two different voltage sources. When there are no "mechanical" boundary forces acting in the longitudinal direction, it is shown that the system with certain parameter combinations is not uniformly strongly stabilizable by the $B^*-$type feedback controller, which is the total current accumulated at the electrodes for the piezoelectric layers. However, as the magnetic effects are ignored (electrostatic assumption), the closed-loop system with all mechanical feedback controllers is shown to be uniformly exponentially stable.

math.AP↗

arXiv2016

Modeling and related results for current-actuated piezoelectric beams by including magnetic effects

Piezo-electric material can be controlled with current as the electrical variable, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. Besides the consideration of current control, there are several new aspects to the model here. Most significantly, magnetic effects are included. For the electromagnetic part of the model, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Two sets of decoupled system of partial differential equations are obtained; one for stretching motion and another one for bending motion. Hamilton's principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a charge (or current) source at the electrodes. Current or charge controllers at the electrodes can only control the stretching motion. Attention is therefore focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb-type gauge condition which is widely used in the electromagnetic theory is used here. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike the voltage or charge actuation, a bounded control operator in the natural energy space is obtained in the current actuation case. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models.

math.AP↗

arXiv2016

Modeling and stabilization results for a charge or current-actuated active constrained layer (ACL) beam model with the electrostatic assumption

An infinite dimensional model for a three-layer active constrained layer (ACL) beam model, consisting of a piezoelectric elastic layer at the top and an elastic host layer at the bottom constraining a viscoelastic layer in the middle, is obtained for clamped-free boundary conditions by using a thorough variational approach. The Rao-Nakra thin compliant layer approximation is adopted to model the sandwich structure, and the electrostatic approach (magnetic effects are ignored) is assumed for the piezoelectric layer. Instead of the voltage actuation of the piezoelectric layer, the piezoelectric layer is proposed to be activated by a charge (or current) source. We show that, the closed-loop system with all mechanical feedback is shown to be uniformly exponentially stable. Our result is the outcome of the compact perturbation argument and a unique continuation result for the spectral problem which relies on the multipliers method. Finally, the modeling methodology of the paper is generalized to the multilayer ACL beams, and the uniform exponential stabilizability result is established analogously.

math.AP↗

arXiv2015

Modeling and semigroup formulation of charge or current-controlled active constrained layer (ACL) beams; electrostatic, quasi-static, and fully-dynamic assumptions

A three-layer active constrained layer (ACL) beam model, consisting of a piezoelectric elastic layer, a stiff layer, and a constrained viscoelastic layer, is obtained for cantilevered boundary conditions by using the reduced Rao-Nakra sandwich beam assumptions through a consistent variational approach. The Rao-Nakra sandwich beam assumptions keeps the longitudinal and rotational inertia terms. We consider electrostatic, quasi-static and fully dynamic assumptions due to Maxwell's equations. For that reason, we first include all magnetic effects for the piezoelectric layer. Two PDE models are obtained; one for the charge-controlled case and one for the current-controlled case. These two cases are considered separately since the underlying control operators are very different in nature. For both cases, the semigroup formulations are presented, and the corresponding Cauchy problems are shown to be well- posed in the natural energy space.

math.AP↗

arXiv2015

Further stabilization results for voltage-actuated piezoelectric beams with magnetic effects

It is well known that magnetic energy of the piezoelectric beam is relatively small, and it does not change the overall dynamics. Therefore, the models, relying on electrostatic or quasi-static approaches, completely ignore the magnetic energy stored/produced in the beam. A single piezoelectric beam model without the magnetic effects is known to be exactly observable and exponentially stabilizable in the energy space. However, the model with the magnetic effects is proved to be not exactly observable / exponentially stabilizable in the energy space for almost all choices of material parameters. Moreover, even strong stability is not achievable for many values of the material parameters. In this paper, it is shown that the uncontrolled system is exactly observable in a space larger than the energy space. Then, by using a $B^*-$type feedback controller, explicit polynomial decay estimates are obtained for more regular initial data. Unlike the classical counterparts, this choice of feedback corresponds to the current flowing through the electrodes, and it matches better with the physics of the model. The results obtained in this manuscript have direct implications on the controllability/stabilizability of smart structures such as elastic beams/plates with piezoelectric patches and the active constrained layer (ACL) damped beams/plates.

math.AP↗

arXiv2013

Uniform stabilization of a multilayer Rao-Nakra sandwich beam

We consider the problem of boundary feedback stabilization of a multilayer Rao-Nakra sandwich beam. We show that the eigenfunctions of the decoupled system form a Riesz basis. This allows us to deduce that the decoupled system is exponentially stable. Since the coupling terms are compact, the exponential stability of the coupled system follows from the strong stability of the coupled system, which is proved using a unique continuation result for the overdetermined homogenous system in the case of zero feedback.

math.AP↗