55 research outputs found
Unbounded Viscosity Solutions of Hybrid Control Systems
We study a hybrid control system in which both discrete and continuous
controls are involved. The discrete controls act on the system at a given set
interface. The state of the system is changed discontinuously when the
trajectory hits predefined sets, namely, an autonomous jump set or a
controlled jump set where controller can choose to jump or not. At each
jump, trajectory can move to a different Euclidean space. We allow the cost
functionals to be unbounded with certain growth and hence the corresponding
value function can be unbounded. We characterize the value function as the
unique viscosity solution of the associated quasivariational inequality in a
suitable function class. We also consider the evolutionary, finite horizon
hybrid control problem with similar model and prove that the value function is
the unique viscosity solution in the continuous function class while allowing
cost functionals as well as the dynamics to be unbounded
Observers for compressible Navier-Stokes equation
We consider a multi-dimensional model of a compressible fluid in a bounded
domain. We want to estimate the density and velocity of the fluid, based on the
observations for only velocity. We build an observer exploiting the symmetries
of the fluid dynamics laws. Our main result is that for the linearised system
with full observations of the velocity field, we can find an observer which
converges to the true state of the system at any desired convergence rate for
finitely many but arbitrarily large number of Fourier modes. Our
one-dimensional numerical results corroborate the results for the linearised,
fully observed system, and also show similar convergence for the full nonlinear
system and also for the case when the velocity field is observed only over a
subdomain
Hybrid control systems and viscosity solutions
We investigate a model of hybrid control system in which both discrete and continuous controls are involved. In this general model, discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set A or a controlled jump set C where the controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. We prove the continuity of the associated value function V with respect to the initial point. Using the dynamic programming principle satisfied by V, we derive a quasi-variational inequality satisfied by V in the viscosity sense. We characterize the value function V as the unique viscosity solution of the quasi-variational inequality by the comparison principle method
Positive solution branch for elliptic problems with critical indefinite nonlinearity
In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely -Îu=λu+h(x)un+2/n-2 in a smooth domain bounded (respectively, unbounded) ΩâRn, n>4, for λâ„0. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue λ1(Ω) (respectively, the bottom of the essential spectrum)
Studying the Issues of Language Connections of Uzbek and Tadjik Languages
In this paper researched the most important elements of language contacts of Uzbek and Tajik, and although Tajik and Uzbek languages, some issues of this bilingualism between two languages.Keywords: language contacts, bilingualism, culture, ethnical group, lexical system
Self-propelled motion of a rigid body inside a density dependent incompressible fluid
This paper is devoted to the existence of a weak solution to a system describing a self-propelled motion of a rigid body in a viscous fluid in the whole R 3. The fluid is modelled by the incompressible nonhomogeneous Navier-Stokes system with a nonnegative density. The motion of the rigid body is described by the balance of linear and angular momentum. We consider the case where slip is allowed at the fluid-solid interface through Navier condition and prove the global existence of a weak solution
Some elliptic semilinear indefinite problems on R<SUP>N</SUP>
This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem âÎu=λf(x,u), uâD1,2(RN). The function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour
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