Conditions For Existence Of Solutions Research Articles

Nonlinear effects of colored nonstationary noise: Exact results

The master equation is derived for random systems under nonlinear time-dependent conditions. The (non-Markov) process is of such a type that with a time-dependent state transformation the dynamics can be modelled by a nonlinear but drift-free Langevin equation. The focus is on the statistical content of resulting master equation. The existence of stationary solutions and the quality of approximative results is discussed.

Application of Altman’s contractor techniques to nonlinear integral equations for Green’s functions of augmented quantum field theory

Altman’s theory of contractors and contractor directions is applied to equations for connected irreducible Green’s functions of Klauder’s augmented quantum field theory in the φ4 model, and sufficient conditions for existence of solutions are discussed.

On the existence of the solution of an optimal control problem for time-lag systems: PMM vol. 43, no. 1, 1979, pp. 17–23

The existence is investigated of the solution of an optimal control problem with a performance index specified as an integral quadratic form of coordinates and controls of a plant described by a system of linear differential equations with lagging argument A connection is established between the given problem and optimal problems of a special kind, for which the solution existence conditions have been investigated in detail.

Minimum principles for the stress field in an elastic-perfectly plastic body

Two minimum principles: for the residual stress field and for the stress field, corresponding to the boundary value problem for a general elastic-perfectly plastic solid are derived. The necessary and sufficient condition for existence of solution is given. The uniqueness of the stress field is proved.

Minimum principles and uniqueness of strain for an elastic-plastic body

The boundary value problem is presented in terms of the generalized stress and strain fields. Two minimum principles: for the generalized residual stress field and for the generalized stress field are derived. The necessary and sufficient condition for existence of solution is given. The uniqueness of the generalized stress field is proved. The regular elastic-plastic material, for which the solution always exists and is unique, is proposed.

Some properties of extremal solutions of zero-sum differential games

Necessary conditions for infinite-time linear-quadratic two-player differential games with perfect state information are examined in the light of Willems' work on the algebraic Riccati equation [1]. Necessary conditions for the nonzero-sum game do not always reduce precisely to those of the zero-sum game, when the weighting parameters of the cost index are taken as those of a zero-sum game. Formulation of zero-sum game problems in extended L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -spaces reveals the existence of nonstabilizing but infimal controls, and explicit frequency-domain conditions for existence of solutions and stabilizing solutions are established through the algebraic Riccati equation (ARE).

Minimization Problems and Linear and Nonlinear Spline Functions. I: Existence

The nonconvex problem of minimum curvature is considered in detail and sufficient conditions for existence of solutions as well as characterizations are presented. The $L^\infty $ minimization problem of least norm, involving a nonsingular linear differential operator and linear inequality constraints, is considered and an existence theorem is presented. A thorough analysis is given for the problem of finding an element in the set $TU$ which is closest to a given element y, where T is a linear mapping of a Banach space X into a complete metric linear space $Y,U$ is a convex subset of X and y is an element of Y. These three classes of results, stated in reverse order to that in which they appear, are seen to be consequences of certain general theorems stated at the outset.

Nonlinear Hydromagnetic Waves in a Collisionless Plasma with Isothermal Electron Pressure. I. Wave Train Propagating across a Magnetic Field

Nonlinear hydromagnetic waves in a hot collisionless plasma are investigated, on the basis of a two-fluid model of plasma composed of cold ions and isothermal electrons. In this paper (Part I), wave trains propagating across a magnetic field are studied analytically. An existence condition for solutions is exactly obtained, and the typical wave profiles are given graphically. These waves are qualitatively similar to those in a cold plasma. The width of the wave is of the order of the geometric mean between the Larmor radii of the electron and of the ion. As limiting cases, they include two special solutions which have already known, that is, (i) solitary wave solutions in a hot plasma, (ii) wave train solutions in a cold plasma.

Application of Liapunov theory to boundary value problems. II

The second order vector differential equation x = f ( t , x , x ′ ) x = f(t,x,x’) is considered. Liapunov conditions for existence of solutions to boundary value problems are obtained strictly in terms of the function f, extending the results of George and Sutton [1].

Proof of an Existence Condition for Solutions of the Unitarity System of Equations

A previously established set of conditions for the existence of solutions of a nonlinear system, expressing the unitarity of n × n symmetric matrices, is proven by means of the homotopy invariance theorem.

Liapunov fundtions and global existence without uniqueness

In a recent paper J. Kato and A. Strauss character- ized the global existence of solutions of ani ordinary differenitial equation in terms of Liapunov functions in which they assumed the right hand side of the differential equatioln is locally Lipschitz. In the present paper a characterization of global existence of an ordinary differential equation is found in which the right hand side is mierely continiuous. The construction of the Liapunov functions depend heavily upon the properties of solution funnels duLe to the nonuniiqueness of solutions. 1. Introduction. In most of the theory dealing with the construction of Liapunov functions for the ordinary differential equation (E) : = f(t ), one assumnes that f is locally Lipschitz in order to obtain a locally Lipschitz Liapunov function. In this paper we consider (E) under the assumption that f:RXRn -Rn is merely coIntiInuous and provide necessary and sufficient conditions for the global existence of solutions of (E) in terms of Liapunov functions whiclh depend upon solution funnels. Kato and Strauss (1 ) have considered this problem assuming f is locally Lipschitz. Their results, as we shall show in an example, do not hold when f is continuous even if we do not require the Liapunov function to be continuous.

Existence theorems for second order boundary value problems

Necessary and sufficient conditions for solution existence to second order boundary value problems, considering second order differential equation involving function of three variables

Contribution to the problem on the analytical construction of regulators

The existence is investigated of the solution of an optimal control problem with a performance index specified as an integral quadratic form of coordinates and controls of a plant described by a system of linear differential equations with lagging argument A connection is established between the given problem and optimal problems of a special kind, for which the solution existence conditions have been investigated in detail.

On a Method for Solving the Balance Equation in Typhoon Region

In order to avoid the truncation errors near the typhoon center, which arise in the course of numerical computation, and to obtain the stream function in the vicinity of typhoon precisely, the balance equation is separated into two parts; the one is corresponding to the circular symmetric field and the other, to the residual field. The stream function corresponding to the circular symmetric field is easily obtained by the gradient wind equation. After the condition for existence of solutions of the balance equation corresponding to the residual field is examined, the point relaxation procedure to solve this equation is proposed. An example of this solving method, applying to the typhoon Harriet, is presented also.

A Geometric Framework for Stabilization by Energy Shaping: Sufficient Conditions for Existence of Solutions

We present a geometric formulation for the energy shaping problem. The central objective is the initiation of a more systematic exploration of energy shaping with the aim of de- termining whether a given system can be stabilized using energy shaping feedback. We investigate the partial differential equations for the kinetic energy shaping problem using the formal theory of partial differential equations. The main contribution is sufficient conditions for integrability of these partial differential equations. We couple these results with the integrability results for potential energy shaping (25). This gives some new avenues for answering key questions in energy shaping that have not been addressed to this point. 1. Introduction. In Brockett's 1977 paper (9) it was observed that there were structural aspects of mechanical systems that made them attractive as a class of control problems. In this paper he mentioned differential geometry as the common mathematical structure between control theory and analytical mechanics. He inves- tigated the Lagrangian and Hamiltonian formulations for mechanical systems and considered the interplay of the mechanical and control theoretic structures. One interesting control problem is the following: given a mechanical system with an unstable equilibrium at a point q0, stabilize the system using feedback. One of the recent developments in the stabilization of equilibria is the energy shaping method. The key idea concerns the construction of a feedback for which the closed-loop system possesses the structure of a mechanical system. A feedback so obtained is called an energy shaping feedback and the procedure by which it is obtained is called energy shaping. In the classical notion of energy shaping, the assumed method consists of two stages: shaping the kinetic energy of the system—so-called kinetic energy shaping—and changing the potential energy of the system—so-called potential energy shaping. If such an energy shaping feedback exists, then for stability one has to ensure that the Hessian of the closed-loop potential energy is positive definite. The cart-pendulum, as a mechanical system with one degree of underactuation, is one of the systems that has been stabilized using the energy shaping method (14, 26). The system has the upright equilibria as saddle points and potential energy