scholarly journals Explicit state and output feedback boundary controllers for partial differential equations

2003 ◽  
Vol 13 (2) ◽  
pp. 1-9 ◽  
Author(s):  
Andrey Smyshlyaev ◽  
Miroslav Krstic
Keyword(s):  
Closed Form ◽  
Output Feedback ◽  
Closed Loop ◽  
Parabolic Problems ◽  
Stabilization Problem ◽  
Hyperbolic Partial Differential Equation ◽  
Boundary Stabilization ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Sensor Actuator

In this paper the explicit (closed form) solutions to several application-motivated parabolic problems are presented. The boundary stabilization problem is converted to a problem of solving a specific linear hyperbolic partial differential equation (PDE). This PDE is then solved for several particular cases. Closed loop solutions to the original parabolic problem are also found explicitly. Output feedback problem under boundary measurement is explicitly solved with both anti-collocated and collocated sensor/actuator locations. It is shown how closed form frequency domain compensators based on the closed form observers and controllers can be designed.

Active-Passive Hybrid Constrained Layer for Structural Damping Augmentation

2000 ◽  
Vol 122 (3) ◽  
pp. 254-262 ◽  
Author(s):  
Yanning Liu ◽  
K. W. Wang
Keyword(s):  
Closed Form ◽  
Viscoelastic Material ◽  
Closed Loop ◽  
Equations Of Motion ◽  
Generic Model ◽  
Structural Damping ◽  
Active Materials ◽  
Closed Form Solutions ◽  
Fail Safe ◽  
Active Constrained Layer

A new surface-damping concept with an active-passive hybrid constraining layer (HCL) is proposed to improve the damping performance of traditional active constrained layer (ACL) systems. Instead of using a pure piezoelectric constraining layer, passive and active materials are used together to constrain the viscoelastic material layer. A generic model of the HCL treatment is presented. Nondimensional equations of motion and boundary and connecting conditions are derived. The closed-form solutions to the equations are developed and analyzed. Tabletop tests are also performed to verify the feasibility of the new damping concept. It is shown that by properly selecting a passive constraining material and assigning appropriate lengths for the active and passive constraining parts, HCL can outperform a system with a pure active PZT coversheet, both in terms of its fail-safe ability and closed-loop damping performance. [S0739-3717(00)01503-8]

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

scholarly journals Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation

2022 ◽  
Author(s):  
Hua-Cheng Zhou ◽  
Ze-Hao Wu ◽  
Bao-Zhu Guo ◽  
Yangquan Chen
Keyword(s):  
Output Feedback ◽  
Disturbance Rejection ◽  
State Feedback ◽  
Closed Loop ◽  
External Disturbance ◽  
Fractional Diffusion ◽  
Loop System ◽  
Boundary Stabilization ◽  
Closed Loop System ◽  
Diffusion Wave

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in [Nonlinear Dynam., 38(2004), 339-354] where all results were verified by simulations only.

TRANSIENT ANALYSIS OF IMMIGRATION BIRTH–DEATH PROCESSES WITH TOTAL CATASTROPHES

2003 ◽  
Vol 17 (1) ◽  
pp. 83-106 ◽  
Author(s):  
Xiuli Chao ◽  
Yuxi Zheng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Closed Form ◽  
Stochastic Systems ◽  
Equilibrium Distribution ◽  
Partial Differential ◽  
Equilibrium Behavior ◽  
Closed Form Solutions ◽  
Population Process ◽  
Transient Solutions

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

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