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Block Backstepping for Isotachic Hyperbolic PDEs and Multilayer Timoshenko Beams

Guangwei Chen, Rafael Vazquez, Junfei Qiao, Miroslav Krstic

TL;DR

This work addresses rapid stabilization of $N$-layer Timoshenko beams with anti-damping and anti-stiffness at the uncontrolled boundary by extending backstepping to isotachic hyperbolic PIDE-ODE systems. A Riemann-type transformation converts the multilayer beam dynamics into a 1-D hyperbolic PIDE-ODE with block structure, enabling a block backstepping design that handles identical transport speeds within blocks. The authors derive a boundary control law with gain kernels determined from nonlinear kernel PDEs and prove exponential stability with an arbitrarily large decay rate by placing $E_1=A+Boldsymbol{ abla}(0)$ appropriately. Numerical simulations for a two-layer beam validate rapid stabilization and demonstrate the method’s capability to manage complex spatial-temporal dynamics while preserving inter-layer coupling.

Abstract

In this paper, we investigate the rapid stabilization of N-layer Timoshenko composite beams with anti-damping and anti-stiffness at the uncontrolled boundaries. The problem of stabilization for a two-layer composite beam has been previously studied by transforming the model into a 1-D hyperbolic PIDE-ODE form and then applying backstepping to this new system. In principle this approach is generalizable to any number of layers. However, when some of the layers have the same physical properties (as e.g. in lamination of repeated layers), the approach leads to isotachic hyperbolic PDEs (i.e. where some states have the same transport speed). This particular yet physical and interesting case has not received much attention beyond a few remarks in the early hyperbolic design. Thus, this work starts by extending the theory of backstepping control of (m + n) hyperbolic PIDEs and m ODEs to blocks of isotachic states, leading to a block backstepping design. Then, returning to multilayer Timoshenko beams, the Riemann transformation is used to transform the states of N-layer Timoshenko beams into a 1-D hyperbolic PIDE-ODE system. The block backstepping method is then applied to this model, obtaining closed-loop stability of the origin in the L2 sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Finally, numerical simulations are presented corroborating the theoretical developments.

Block Backstepping for Isotachic Hyperbolic PDEs and Multilayer Timoshenko Beams

TL;DR

This work addresses rapid stabilization of -layer Timoshenko beams with anti-damping and anti-stiffness at the uncontrolled boundary by extending backstepping to isotachic hyperbolic PIDE-ODE systems. A Riemann-type transformation converts the multilayer beam dynamics into a 1-D hyperbolic PIDE-ODE with block structure, enabling a block backstepping design that handles identical transport speeds within blocks. The authors derive a boundary control law with gain kernels determined from nonlinear kernel PDEs and prove exponential stability with an arbitrarily large decay rate by placing appropriately. Numerical simulations for a two-layer beam validate rapid stabilization and demonstrate the method’s capability to manage complex spatial-temporal dynamics while preserving inter-layer coupling.

Abstract

In this paper, we investigate the rapid stabilization of N-layer Timoshenko composite beams with anti-damping and anti-stiffness at the uncontrolled boundaries. The problem of stabilization for a two-layer composite beam has been previously studied by transforming the model into a 1-D hyperbolic PIDE-ODE form and then applying backstepping to this new system. In principle this approach is generalizable to any number of layers. However, when some of the layers have the same physical properties (as e.g. in lamination of repeated layers), the approach leads to isotachic hyperbolic PDEs (i.e. where some states have the same transport speed). This particular yet physical and interesting case has not received much attention beyond a few remarks in the early hyperbolic design. Thus, this work starts by extending the theory of backstepping control of (m + n) hyperbolic PIDEs and m ODEs to blocks of isotachic states, leading to a block backstepping design. Then, returning to multilayer Timoshenko beams, the Riemann transformation is used to transform the states of N-layer Timoshenko beams into a 1-D hyperbolic PIDE-ODE system. The block backstepping method is then applied to this model, obtaining closed-loop stability of the origin in the L2 sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Finally, numerical simulations are presented corroborating the theoretical developments.
Paper Structure (14 sections, 3 theorems, 53 equations, 5 figures)

This paper contains 14 sections, 3 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Controller design
  4. Block transformation for isotachic states
  5. Boundary control law and stability result
  6. Controller Analysis
  7. Target system
  8. Backstepping transformation
  9. Stability and Analysis of Closed Loop
  10. A semi-explicit solution for the target system
  11. Lyapunov-based stability analysis of target system
  12. Rapid Stabilization of N-layer Timoshenko Beam
  13. Numerical simulation
  14. Concluding remarks

Key Result

theorem 1

Consider system (plant_eq1_Hyperbolic)--(BDm_e_hyperbolic), with initial conditions $Z_{0},Y_{0} \in L^2(0,1)$, $X_{0} \in L^2$ under the control law (eqn-controlaw_new). For all $C_2>0$ there exists gains $K(1,y)$, $L(1,y)$ and $\Phi(1)$ such that (plant_eq1_Hyperbolic)--(BDm_e_hyperbolic) has a so

Figures (5)

  • Figure 1: Illustration of multilayer Timoshenko beam system
  • Figure 2: The procedure of power series method
  • Figure 3: Solutions of Timoshenko gain kernels $K_{ij}(1,y),L_{ij}(1,y), 1\le i\le4,1\le j\le4$ (from left to right).
  • Figure 4: Evolution of open-loop Timoshenko states $v_1(x,t),\theta_1(x,t), v_2(x,t),\theta_2(x,t)$ (from left to right).
  • Figure 5: Evolution of closed-loop Timoshenko states $v_1(x,t),\theta_1(x,t), v_2(x,t),\theta_2(x,t)$ (from left to right).

Theorems & Definitions (3)

  • theorem 1
  • theorem 2
  • theorem 3