Superfield Quantization in the Sp(2)-Covariant Formalism

We generalize the rules for the superfield Sp(2)-covariant quantization of arbitrary gauge theories to the case of gauge fixing by the generating equations for the gauge functional. We consider possible realizations of the extended antibrackets and show that only one of the realizations is consistent with the extended BRST symmetry transformations in the form of the supertranslations along the Grassmann coordinates of a superspace.     

A method for solving boundary value problems for nonlinear control systems in the class of discrete controls

We suggest an algorithm for constructing discrete control functions. This algorithm is sufficiently convenient for numerical implementation and, for a wide class of nonlinear systems of ordinary differential equations, provides the passage from the initial state into an arbitrary given or arbitrarily small neighborhood of a given terminal state. We obtain a constructive criterion for the choice of terminal states and discretization steps for which the passage is possible with regard of the constraints imposed on the control and the phase coordinates. We consider an interorbital flight problem, for which we carry out numerical simulation.     

On the strong solutions of a regularized model of a nonlinear visco-elastic medium

We consider the initial boundary-value problem for the system of equations describing the motion of a nonlinear visco-elastic medium with memory along the trajectories of the velocity field; the system in question is a generalization of the system of Navier-Stokes equations. We establish existence and uniqueness theorems for strong solutions containing higher derivatives that are square-integrable in the plane case.     

Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions

This paper presents extensions of some nonexistence results for elliptic systems with dynamical boundary conditions involving the time-derivatives of integer orders to the case of noninteger order. In particular, we consider a system of Poisson’s equations with time-fractional derivatives of order less than one in the boundary conditions and specify the thresholds of the nonlinearities which lead to the absence of global solutions. The fractional derivatives here are meant in the Riemann-Liouville sense (or in the Caputo sense). We also present necessary conditions for the existence of local solutions.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.     

On Solvability of an Initial-Boundary Value Problem for a Viscoelasticity Model with Fractional Derivatives

We establish the existence and uniqueness (the latter only in the plane case) of a weak solution to an initial-boundary value problem for the system of the equations of motion of a viscoelastic fluid, namely, for the anti-Zener model whose constitutive law contains fractional derivatives. We use the approximation of this problem by a sequence of regularized Navier–Stokes systems and passage to the limit.     

Study of the passage from a dissipative to a conservative state in two-dimensional nonlinear systems of ordinary differential equations

We consider an approximate model of a space pendulum described by a system of two equations. We construct a two-dimensional diagram in the parameter space showing the passage from a dissipative to a conservative state in this system. We show that the Poincaré map of this system can be represented as the square of some other map; this fact is used to explain some specific features of the structure of the bifurcation diagram of the original map.     

Structure of weak laminar hydraulic jumps in single layer and two layer fluids

We consider the occurrence of hydraulic jumps in near critical single layer and two layer flows under the assumption that viscous effects are confined to a thin laminar boundary layer adjacent to the solid boundary. In the limit of large Reynolds number this leads to a structure problem formed by the classical triple deck equations supplemented with a novel nonlinear coupling condition which allows for the passage through the critical state. In the case of positive hydraulic jumps this passage is achieved by the local thickening of the boundary layer which acts as a viscous hump. Conversely, the pressure drop at the wall associated with negative hydraulic jumps causes the boundary layer to decrease locally thereby forming a local indentation required for the Froude number to pass through one in this case.     

Boundary control by displacements at one endpoint of an inhomogeneous rod for fixed other endpoint in the case of coinciding times of passage of the wave through either of the inhomogeneity parts

In the present paper, we consider a boundary control by displacements at one endpoint of an inhomogeneous rod that has two parts of different densities and elasticities with fixed other endpoint for the case in which the times of passage of the wave through each of these inhomogeneity parts coincide. We find a closed analytic form of the boundary control by displacements bringing the stick from the original rest state into a given terminal state specified by given terminal displacement and velocity.     

Weakly invariant sets of hybrid systems

We consider a mathematical model of a hybrid system in which the continuous dynamics generated at any point in time by one of a given finite family of continuous systems alternates with discrete operations commanding either an instantaneous switching from one system to another, or an instantaneous passage from current coordinates to some other coordinates, or both operations simultaneously. As a special case, we consider a model of a linear switching system. For a hybrid system, we introduce the notion of a weakly invariant set and analyze its structure. We obtain a representation of a weakly invariant set as a union of sets of simpler structure. For the latter sets, we introduce special value functions, for which we obtain expressions by methods of convex analysis. For the same functions, we derive equations of the Hamilton-Jacobi-Bellman type, which permit one to pass from the problem of constructing weakly invariant sets to the control synthesis problem for a hybrid system.     

Optimal Control of State Constrained Integral Equations

We consider the optimal control problem of a class of integral equations with initial and final state constraints, as well as running state constraints. We prove Pontryagin’s principle, and study the continuity of the optimal control and of the measure associated with first order state constraints. We also establish the Lipschitz continuity of these two functions of time for problems with only first order state constraints.     

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.     

Rejection rules in theM/G/1 queue

We consider aM/G/1 queue modified such that an arriving customer may be totally or partially rejected depending on a r.v. (the barricade) describing his impatience and on the state of the system. Three main variants of this scheme are studied. The steady-state distribution is expressed in terms of Volterra equations and the relation to storage processes, dams and queues with state-dependent Poisson arrival rate is discussed. For exponential service times, we further find the busy period Laplace transform in the case of a deterministic barricade, whereas for exponential barricade it is shown by a coupling argument that the busy period can be identified with a first passage time in an associated birth-death process.     

On the reduction of third-order vector linear system to a form with Isidori relative order

We consider a special case of a linear vector dynamical system. We study the possibility of reducing system outputs to a form with Isidori relative order by a nonsingular transformation.     

Optimal Trade Execution Under Stochastic Volatility and Liquidity

Abstract

We study the problem of optimally liquidating a financial position in a discrete-time model with stochastic volatility and liquidity. We consider the three cases where the objective is to minimize the expectation, an expected exponential or a mean-variance criterion of the implementation cost. In the first case, the optimal solution can be fully characterized by a forward-backward system of stochastic equations depending on conditional expectations of future liquidity. In the other two cases, we derive Bellman equations from which the optimal solutions can be obtained numerically by discretizing the control space. In all three cases, we compute optimal strategies for different simulated realizations of prices, volatility and liquidity and compare the outcomes to the ones produced by the deterministic strategies of Bertsimas and Lo (1998; Optimal control of execution costs. Journal of Financial Markets, 1, 1–50) and Almgren and Chriss (2001; Optimal execution of portfolio transactions. Journal of Risk, 3, 5–33).     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

The stabilization of program motions of controlled nonlinear mechanical systems

We consider a controlled nonlinear mechanical system described by the Lagrange equations. We determine the control forcesQ ₁ and the restrictions for the perturbationsQ ₂ acting on the mechanical system which allow to guarantee the asymptotic stability of the program motion of the system. We solve the problem of stabilization by the direct Lyapunov's method and the method of limiting functions and systems. In this case we can use the Lyapunov's functions having nonpositive derivatives. The following examples are considered: stabilization of program motions of mathematical pendulum with moving point of suspension and stabilization of program motions of rigid body with fixed point.     

A probabilistic theory of random maps

We present a probabilistic theory of random maps with discrete time and continuous state. The forward and backward Kolmogorov equations as well as the FPK equation governing the evolution of the probability density function of the system are derived. The moment equations of arbitrary order are derived, and the reliability and first passage time problem are also studied. Examples are presented to demonstrate the application of the theoretical development. Numerical solutions including the time histories of moment evolution, steady state probability density function, reliability and first passage time probability density function for time discrete random maps are included. The present work compliments the existing theory of continuous time stochastic processes.     

Optimal control of impulsive hybrid systems

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms.     

Symmetries in equations of incompressible viscoelastic Maxwell medium<Superscript>*</Superscript>

We consider unsteady flows of incompressible viscoelastic Maxwell medium with upper, low, and Jaumann convective derivatives in the rheological constitutive law. We give characteristics of a system of equations that describe a three-dimensional motion of such a medium for all three types of convective derivative. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. We study group properties of this system in the two-dimensional case. On this basis, we choose submodels of the Maxwell model that can be reduced to hyperbolic ones. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. We also present concrete examples of invariant solutions.