6 research outputs found
Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space
In this article we are interested in the boundary stabilization in finite
time of one-dimensional linear hyperbolic balance laws with coefficients
depending on time and space. We extend the so called "backstepping method" by
introducing appropriate time-dependent integral transformations in order to map
our initial system to a new one which has desired stability properties. The
kernels of the integral transformations involved are solutions to non standard
multi-dimensional hyperbolic PDEs, where the time dependence introduces several
new difficulties in the treatment of their well-posedness. This work
generalizes previous results of the literature, where only time-independent
systems were considered
In-Domain Control of Partial Differential Equations
RĂSUMĂ Cette thĂšse porte sur la commande des systĂšmes Ă dimension infinie dĂ©crit par les Ă©qua-tions aux dĂ©rivĂ©es partielles (EDP). La commande dâEDP peut ĂȘtre divisĂ©e approximative-ment en deux catĂ©gories en fonction de lâemplacement des actionneurs: la commande Ă la frontiĂšre, oĂč les actionnements sont appliquĂ©s Ă la frontiĂšre des systĂšmes dâEDP, et la com-mande dans le domaine, oĂč les actionneurs pĂ©nĂštrent Ă lâintĂ©rieur du domaine des systĂšmes dâEDP. Dans cette thĂšse, nous Ă©tudierons la commande dans le domaine de lâĂ©quation dâEuler-Bernoulli, de lâĂ©quation de Fisher, lâĂ©quation de Chafee-Infante et de lâĂ©quation de Burgers. LâĂ©quation dâEuler-Bernoulli est un modĂšle classique dâEDP linĂ©aire dĂ©crivant la flexion pure des structures flexibles. LâĂ©quation de Fisher et lâĂ©quation de Chafee-Infante sont des EDP paraboliques semi-linĂ©aires, qui peuvent ĂȘtre utilisĂ©es pour modĂ©liser certains phĂ©nomĂšnes physiques, chimiques ou biologiques. LâĂ©quation de Burgers peut ĂȘtre considĂ©rĂ©e comme une simplification dâĂ©quations de Navier-Stokes en mĂ©canique des fluides, en dynamique des gaz, en fluiditĂ© de la circulation, etc. Ces systĂšmes jouent des rĂŽles trĂšs importants en mathĂ©ma-tiques, en physique et dans dâautres domaines.
Dans cette thĂšse, de nouvelles mĂ©thodes qui se basent sur la dynamique des zĂ©ros et le compensateur dynamique ont Ă©tĂ© dĂ©veloppĂ©es pour la conception et lâimplĂ©mentation de lois de commande pour la commande des EDP avec des actionnements dans le domaine. Tout dâabord, nous Ă©tudions le contrĂŽle de lâĂ©quation dâEuler-Bernoulli avec plusieurs actionneurs internes. Lâinverse de la dynamique des zĂ©ros a Ă©tĂ© utilisĂ© dans la conception de la loi de commande, ce qui permet de suivre la trajectoire prescrit souhaitĂ©e. Afin de concevoir la trajectoire souhaitĂ©e, la fonction de Green est utilisĂ©e pour dĂ©terminer la commande sta-tique. La planification de mouvement est gĂ©nĂ©rĂ©e par des contrĂŽleurs dynamiques basĂ©s sur la mĂ©thode de platitude diËĂ©rentielle. Pour les Ă©quations paraboliques non linĂ©aires, la dy-namique des zĂ©ros est rĂ©gie par une EDP non linĂ©aire. Par consĂ©quent, nous avons recours Ă la mĂ©thode de dĂ©composition dâAdomian (ADM) pour gĂ©nĂ©rer la commande dynamique afin de suivre les rĂ©fĂ©rences dĂ©sirĂ©es. Dans le cas de lâĂ©quation de Burgers, un compensateur dynamique a Ă©tĂ© utilisĂ©. Pour obtenir la stabilitĂ© globale de lâĂ©quation de Burgers contrĂŽlĂ©e, une rĂ©troaction non linĂ©aire a Ă©tĂ© appliquĂ©e Ă la frontiĂšre. La mĂ©thode dâADM et la platitude ont Ă©tĂ© utilisĂ©es dans lâimplĂ©mentation du compensateur dynamique.----------ABSTRACT
This thesis addresses in-domain control of partial diËerential equation (PDE) systems. PDE control can in general be classified into two categories according to the location of the ac-tuators: boundary control, where the actuators are assigned to the boundary of the PDE systems, and in-domain control, where the actuation penetrates inside the domain of the PDE systems. This thesis investigates the in-domain control of some well-known PDEs, including the Euler-Bernoulli equation, the Fisherâs equation, the Chafee-Infante equation, and Burgersâ equation. Euler-Bernoulli equation is a classical linear PDE used to describe the pure bending of flexible structures. Fisherâs equation and the Chafee-Infante equation are semi-linear parabolic PDEs that can be used to model physical, chemical, and biolog-ical phenomena. Burgersâ equation can be viewed as simplified Navier-Stokes equations in lower dimensions in applied mathematics, and it has been widely adopted in fluid mechan-ics, gas dynamics, traĂżc flow modeling, etc. These PDE systems play important roles in mathematics, physics, and other fields.
In this work, in-domain control of linear and semi-linear parabolic equations are treated based on dynamic compensators. First, we consider the in-domain control of an Euler-Bernoulli equation with multiple internal actuators. The method of zero dynamics inverse is adopted to derive the in-domain control to allow an asymptotic tracking of the prescribed desired outputs. A linear proportional boundary feedback control is employed to stabilize the Euler-Bernoulli equation around its zero dynamics. To design the desired trajectory, the Greenâs function is employed to determine the static control, and then motion planning is generated by dynamic control based on diËerential flatness. For the semi-linear parabolic equations, zero dynamics are governed by nonlinear PDEs. Therefore, the implementation of the in-domain control of linear PDEs cannot be directly applied. We resort then to the Adomian decomposition method (ADM) to implement the dynamic control in order to track the desired set-points. Finally, the in-domain control of a Burgersâ equation is addressed based on dynamic compensator. A nonlinear boundary feedback control is used to achieve the global stability of the controlled Burgersâ equation, and the ADM as well as the flatness are used in the implementation of the proposed in-domain control scheme