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 $A$ or a controlled jump set $C$ 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 -Δu=λu+h(x)un+2/n-2 in a smooth domain bounded (respectively, unbounded) Ω⊆Rn, n>4, for λ≥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 −Δu=λf(x,u), u∈D1,2(RN). The function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour