Global existence for a quasi-linear evolution equation with a non-convex energy

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

     

Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo-Martin-Schwinger and Wigner-Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov-Kubo-Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the SU(2) algebra of observables in the simplest model of an ideal quantum spin paramagnet.     

Determination of the right-hand side of the Navier-Stokes system of equations and inverse problems for the thermal convection equations

Inverse problem for an evolution equation with a quadratic nonlinearity in the Hilbert space is considered. The problem is, given the values of certain functionals of the solution, to find at each point in time the right-hand side that is a linear combination of those functionals. Sufficient conditions for the nonlocal (in time) existence of a solution (on the whole time interval) are established. An application to the inverse problems for the three-dimensional thermal convection equations of viscous incompressible fluid is considered. Unique nonlocal (in terms of time) solvability of the problem of determining the density of heat sources under the regularity condition of the initial data and sufficiently large dimension of the observation space is proved.     

Distribution of eigenvalues for the discontinuous boundary-value problem with functional-manypoint conditions

In this study, we investigate the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also a point of discontinuity, a finite number of internal points and abstract linear functionals. So our problem is not a pure boundary-value one. We single out a class of linear functionals and find simple algebraic conditions on the coefficients which guarantee the existence of an infinite number of eigenvalues. Also, the asymptotic formulas for the eigenvalues are found. The results obtained in this paper are new, even in the case of boundary conditions either without internal points or without linear functionals.     

A regularity result for a minimum problem in orlicz-sobolev spaces with applications in the study of the dirichlet problem for the operator of hencky-nadai theory

The existence and uniqueness of the solution to the problem of minimum for functionals generated by N- functions are obtained in Orlicz-Sobolev spaces. Applications for some functionals dealing with Hencky theory are given.     

Shape Methods for the Transmission Problem with a Single Measurement

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

Uniqueness of unbounded viscosity solutions for impulse control problem

We study here the impulse control problem in infinite as well as finite horizon. We allow the cost functionals and dynamics to be unbounded and hence the value function can possibly be unbounded. We prove that the value function is the unique viscosity solution in a suitable subclass of continuous functions, of the associated quasivariational inequality. Our uniqueness proof for the infinite horizon problem uses stopping time problem and for the finite horizon problem, comparison method. However, we assume proper growth conditions on the cost functionals and the dynamics.     

Inverse problems for stationary Navier-Stokes systems

An inverse problem for a nonlinear equation in a Hilbert space is considered in which the right-hand side that is a linear combination of given functionals is found from given values of these functionals on the solution. Sufficient conditions for the existence of a solution are established, and the solution set is shown to be homeomorphic to a finite-dimensional compact set. A boundary inverse problem for the three-dimensional thermal convection equations for a viscous incompressible fluid and an inverse magnetohydrodynamics problem are considered as applications.     

Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals

We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton-Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.     

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.     

On a nonlinear heat equation with functional dependence

We reduce the Cauchy problem for a heat equation with the nonlinear right-hand side which depends on some functionals to an equivalent integral equation. Considering mainly Banach spaces of continuous, bounded and exponentially bounded functions, we give some natural sufficient conditions for the existence and uniqueness of solutions to these equations. We give a counterexample which shows that the Lipschitz condition is, in general, insufficient for the Cauchy problem with unbounded data and with functional dependence to guarantee an existence result     

Dual variational principles with deviating argument and discontinuous extremals

We study dual or complementary variational principles for functionals with deviating argument and discontinuous extremals. Local conditions for the existence of a pair of dual extremum principles are given. Results are then applied to the control problem involving linear neutral differential-difference equations.     

Vector measure as controls for explicit nonlinear impulsive system of fed-batch culture

In this paper, we consider an optimal control problem of microbial fermentation process in which glycerol is converted to 1,3-propanediol by Klebsiella pneumoniae in fed-batch culture. During the period of reaction, the variation of pH value is monitored to determine glycerol replenishment quantity, guaranteeing that microorganism can always keep growing fast under enough nutrition. Every time pH value is lower than seven, the quantity of glycerol added is such that pH value returns seven again. Glycerol is poured into reactor at discrete time instant and the quantity is controllable. The problem is to determine for each discrete time instant the glycerol quantity to add and maximize the final concentration of 1,3-propanediol. We present a controlled explicit nonlinear impulsive dynamical system of fed-batch culture with state independent vector measures as controls and study the existence, uniqueness, boundedness, continuous dependence and Gâteaux differentiability of its solution with respect to controls. We then propose a multiple objective programming model and demonstrate the regularity of cost functionals and weak compactness of admissible control set. Finally we discuss the existence of optimal control and implement a hybrid particle swarm optimization algorithm to solve the model optimally. Computational results are presented on a numerical example.     

Parameter identification problem for the equation of motion of membrane with strong viscosity

The parameter identification problem of constant parameters in the equation of membrane with strong viscosity is studied. The problem is formulated by a minimization of quadratic cost functionals by distributive measurements. The existence of optimal parameters and necessary optimality conditions for the parameters are proved.     

Parameter identification problems for an extensible beam equation

Parameter identification problems are studied for extensible beam equations. The problem is formulated by minimizations of quadratic cost functionals by distributive measurements. The existence of optimal parameters and necessary conditions for the parameters are proved.     

Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems

We improve Benci and Rabinowitz's Linking theorem for strongly indefinite functionals, giving estimates for a suitably defined relative Morse index of critical points. Such abstract result is applied to the existence problem of periodic orbits and homoclinic solutions of first order Hamiltonian systems in cases where the Palais-Smale condition does not hold. Received January 27, 1999 / Accepted January 14, 2000 / Published online July 20, 2000     

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Existence of positive solutions for fourth‐order impulsive integral boundary value problems on time scales

This paper is devoted to the existence of positive solutions for a fourth‐order impulsive boundary value problem with integral boundary conditions on time scales. Existence results of at least two and three positive solutions are established via the double fixed point theorem and six functionals fixed point theorem, respectively. Also, an example is given to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

The wavelet transform for Wiener functionals and some applications

The wavelet transform is defined for Wiener functionals. We characterize global and local regularities of Wiener functionals and we give a criterion for the existence and regularity of densities. Such a criterion is applied to diffusion processes and to the solutions to backward stochastic differential equations.