MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

On the global dynamics of attractors for scalar delay equations

A semi-conjugacy from the dynamics of the global attractors for a family of scalar delay differential equations with negative feedback onto the dynamics of a simple system of ordinary differential equations is constructed. The construction and proof are done in an abstract setting, and hence, are valid for a variety of dynamical systems which need not arise from delay equations. The proofs are based on the Conley index theory.

     

On a system of differential equations with operator coefficients and its applications to multidimensional systems of composite type

A mixed problem for a system of differential equations with operator coefficients is considered on an interval. Necessary and sufficient conditions for the existence of at least one solution of the given problem are investigated. It is established that the linear manifold of the solutions of the homogeneous problem is finite-dimensional. The obtained results are applied to multidimensional systems of differential equations of composite type, defined in cylindrical domains.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 25–69, 1992.     

On Some Finite Difference Scheme for Gas Dynamics Equations

A conservative difference scheme with linear dependence of the pressure on the density of gas is proposed for gas dynamics equations. The scheme allows us to simulate 1-D flows inside a cylindrical domain with time-variable cross-sections and guarantees the positive sign of the density function.     

Averaging method applicable to the dynamics of not fully elastic bodies

Using well-known methods, various problems of dynamics are reduced to a standard system of integral differential equations, accounting for heredity effects in the form of multiple integrals according to the theory of Volterra. An averaging scheme is applied to this standard system, converting it to a system of differential equations which are much simpler that the initial equations. A theorem is proved which establishes the similarity of the solutions of these systems.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 5, pp. 940–942, September–October, 1972.     

Effective particle methods for Fisher–Kolmogorov equations: Theory and applications to brain tumor dynamics

Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher–Kolmogorov type equations. These solutions are parametrized by means of a minimal set of time-dependent quantities for which ordinary differential equations ruling their dynamics are found. A comparison of the finite dimensional equations and the dynamics of the full partial differential equation is made showing a very good quantitative agreement with the dynamics of the partial differential equation. We also discuss some implications of our findings for the understanding of the growth progression of certain types of primary brain tumors and discuss possible extensions of our results to related equations arising in different modeling scenarios.     

Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads

In the present paper, the dynamic stability of circular cylindrical shells is investigated; the combined effect of compressive static and periodic axial loads is considered. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration; Lagrange equations are used to reduce the nonlinear partial differential equations to a set of ordinary differential equations. The dynamic stability is investigated using direct numerical simulation and a dichotomic algorithm to find the instability boundaries as the excitation frequency is varied; the effect of geometric imperfections is investigated in detail. The accuracy of the approach is checked by means of comparisons with the literature.     

Numerical analysis of stresses and strains in flexible cylindrical shells with variable rigidity in two directions

The paper analyzes the stress-strain state in a cylindrical shell with variable rigidity in two directions. The shell is analyzed in a geometrically nonlinear framework under different loads and boundary conditions. The proposed approach reduces the nonlinear system of partial differential equations to a sequence of linear systems of ordinary differential equations. The latter are solved by discrete orthogonalization.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 62–67, 1985.     

Propagation of shock waves in an isentropic,nonviscous gas

A new method is considered for constructing the dynamics of motion of a shock front moving with speed close to that of sound in a nonviscous, isentropic gas. The algorithm for constructing the solution reduces to the problem of successively solving systems of ordinary differential equations which provides considerable economy as compared with the known method of difference schemes for computer calculations.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 199–271, 1977.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

Multidimensional characteristic Galerkin methods for hyperbolic systems

We consider characteristic Galerkin methods for the solution of hyperbolic systems of partial differential equations of first order. A new recipe for the construction of approximate evolution operators is given in order to derive consistent methods. With the help of semigroup theory we derive error estimates for classes of characteristic Galerkin methods. The theory is applied to the wave equation and also to the Euler equations of gas dynamics. In the latter case one can show that Fey's genuinely multidimensional method can be reinterpreted as a characteristic Galerkin method. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Conditions for the existence of limit cycles of the second kind for a system of differential equations: II

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.     

Conditions for the existence of limit cycles of the second kind for a system of differential equations: I

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.     

Classification of Differential Invariant Submodels

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.     

Differential Elimination–Completion Algorithms for DAE and PDAE

Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics.
A DAE has differential index r if a minimum of r +1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties.
Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.     

Reduction Methods and Chaos for Quadratic Systems of Differential Equations

We consider systems of differential equations with quadratic nonlinearities having applications for biochemistry and population dynamics, which may have a large dimension n. Due to the complexity of these systems, reduction algorithms play a crucial role in study of their large time behavior. Our approach aims to reduce a large system to a smaller one consisting of m differential equations, where . Under some restrictions (that allow us to separate slow and fast variables in the system) we obtain a new system of differential equations, involving slow variables only. This reduction is feasible from a computational point of view for large n that allows us to investigate sensitivity of dynamics with respect to random variations of parameters. We show that the quadratic systems are capable to generate all kinds of structurally stable dynamics including chaos.     

Global attractivity in monotone and subhomogeneous almost periodic systems

A global attractivity theorem is first proved for a class of skew-product semiflows. Then this result is applied to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations for their global dynamics.     

Hamiltonian Systems near Relative Equilibria

We give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.