A system of partial differential equations describing the spatial oscillations of an Euler-Bernoulli beam with a tip mass is considered. The linear system considered is actuated by two independent controls and separated into a pair of differential equations in a Hilbert space. A feedback control ensuring strong stability of the equilibrium in the sense of Lyapunov is proposed. The proof of the main result is based on the theory of strongly continuous semigroups.
The paper is concerned with the finite-time stabilization of a hybrid PDE-ODE system describing the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is described by the wave equation with a variable coefficient which is an affine function of the curvilinear abscissa along the cable. Using several changes of variables, a backstepping transformation, and a finite-time stable second-order ODE for the dynamics of a conveniently chosen variable, we prove that a global finite-time stabilization occurs for the full system constituted of the platform and the cable. The kernel equations and the finite-time stable ODE are numerically solved in order to compute the nonlinear feedback law, and numerical simulations validating our finite-time stabilization approach are presented.
We present a framework to transform the problem of finding a Lyapunov function of a Chemical Reaction Network (CRN) in concentration coordinates with arbitrary monotone kinetics into finding a common Lyapunov function for a linear parameter varying system in reaction coordinates. Alternative formulations of the proposed Lyapunov function is presented also. This is applied to reinterpret previous results by the authors on Piecewise Linear in Rates Lyapunov functions, and to establish a link with contraction analysis. Persistence and uniqueness of equilibria are discussed also.
Output-based controllers are known to be fragile with respect to model uncertainties. The standard $mathcal{H}_{infty}$-control theory provides a general approach to robust controller design based on the solution of the $mathcal{H}_{infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples.
In current paper, we put forward a reaction-diffusion system for West Nile virus in spatial heterogeneous and time almost periodic environment with free boundaries to investigate the influences of the habitat differences and seasonal variations on the propagation of West Nile virus. The existence, uniqueness and regularity estimates of the global solution for this disease model are given. Focused on the effects of spatial heterogeneity and time almost periodicity, we apply the principal Lyapunov exponent $lambda(t)$ with time $t$ to get the initial infected domain threshold $L^*$ to analyze the long-time dynamical behaviors of the solution for this almost periodic West Nile virus model and give the spreading-vanishing dichotomy regimes of the disease. Especially, we prove that the solution for this West Nile virus model converges to a time almost periodic function locally uniformly for $x$ in $mathbb R$ when the spreading occurs, which is driven by spatial differences and seasonal recurrence. Moreover, the initial disease infected domain and the front expanding rate have momentous impacts on the permanence and extinction of the epidemic disease. Eventually, numerical simulations identify our theoretical results.
We consider the semilinear parabolic equation of normal type connected with the 3D Helmholtz equation with periodic boundary condition. The problem of stabilization to zero of the solution for normal parabolic equation with arbitrary initial condition by starting control is studied. This problem is reduced to establishing three inequalities connected with starting control, one of which has been proved previously. The proof for the other two is given here.