State constrained optimal control problems governed by semilinear equations sup∗

This work deals with necessary conditions of optimality for optimal control problems governed by semilinear differential equations in Hilbert spaces.     

Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Shrödinger equations with complex coefficients

A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrödinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.     

Stability on a cone in terms of two measures for impulsive differential equations with “supremum”

The stability of nonlinear impulsive differential equations with “supremum” is studied. A special type of stability, combining two different measures and a dot product, is defined. The definition is a generalization of several types of stability known in the literature. Razumikhin’s method as well as a comparison method for scalar impulsive ordinary differential equations have been employed.     

Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations

A generalization of the Oberbeck–Boussinesq model consisting of a system of steady state multivalued partial differential equations for incompressible, generalized Newtonian of the p-power type, viscous flow coupled with the nonlinear heat equation is studied in a bounded domain. The existence of a weak solution is proved by combining the surjectivity method for operator inclusions and a fixed point technique.     

On the functional differential equations with “maximums”

The paper deals with the existence-uniqueness problem of the solution of functional differential equations with maximums, which arises in the theory of automatic control. Making use of some previous results of the authors, sufficient conditions are obtained for the existence and uniqueness of the global solution of the initial value problem for these equations.     

On the functional differential equations with “maximums”

The paper deals with an existence - uniqueness problem of the solution for functional differential equations with "maximums", Which arise in the theory of automatic control     

Stochastic differential equations with polar-decomposed L関y measures and applications to stochastic optimization

This paper is concerned with the optimal control of jump-type stochastic differential equations associated with polar-decomposed Lévy measures with the feature of explicit construction on the jump term. The concrete construction is then utilized for analysis of two portfolio optimization problems for financial market models driven by stable-like processes.      

Strict stability in terms of two measures for impulsive differential equations with ‘supremum’

Strict stability for a nonlinear system of impulsive differential equations with ‘supremum’ is defined and studied. Razhumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations are the bases of the main proofs. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied. An example illustrating the usefulness of the obtained sufficient conditions is also included.     

Hamilton–Jacobi equations in space of measures associated with a system of conservation laws

We introduce a class of action integrals defined over probability measure-valued path space. We show that extremal point of such action exits and satisfies a type of compressible Euler equation in a weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton–Jacobi equations, in the space of probability measures, are well-posed.     

Bounded solutions of differential equations with “slow” time

The problem of existence of bounded solutions on the whole number line of a nonlinear system of first order with slow time is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1621–1623, November, 1992.     

L ∞-error estimates of triangular mixed finite element methods for optimal control problems governed by semilinear elliptic equations

We look at L ^∞-error estimates for convex quadratic optimal control problems governed by nonlinear elliptic partial differential equations. In so doing, use is made of mixed finite element methods. The state and costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control, by piecewise constant functions. L ^∞-error estimates of optimal order are derived for a mixed finite element approximation of a semilinear elliptic optimal control problem. Finally, numerical tests are presented which confirm our theoretical results.     

Menshov’s “adjustment theorem” with respect to general measures

We prove Menshov’s theorem in the setting of arbitrary Borel measures.     

An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

Abstract

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation.     

Factorial series connected with the Lambert function,and a problem posed by Ramanujan

Ramanujan’s sequence {y _n } _n=0^∞ defined by is expanded in factorial series derived from a series representing the Lambert W function. As a corollary, it is shown that the sequence {y _n } is completely monotonic.      

“Classical” equations of motion in quantum mechanics with gauge fields

For the Schrödinger and Dirac equations in an external gauge field with symmetry groupSU(2), we construct to any degree of accuracy in powers ofh ^1/2,h0, approximate dynamical states in the form of wave packets—semiclassical trajectory-coherent states. For the quantum expectation values calculated with respect to these semiclassical states we obtain for the operators of the coordinates, momenta, and isospin of the particle Hamiltonian equations of motion that are equivalent (in the relativistic case after transition to the proper time) to Wong's well-known equations for a non-Abelian charge with isospin 1/2.Moscow Engineering Physics Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 41–61, July, 1992.