The construction of asymptotic solutions for quasi-linear partial differential equations with retarded time argument

Ukrainian Mathematical Journal -     

Periodic solutions for higher order differential equations with deviating argument

By using the coincidence degree theory of Mawhin, we study the existence of periodic solutions for higher order differential equations with deviating argument . Some new results on the existence of periodic solutions of the equations are obtained. Meanwhile, an example is given to illustrate our results.     

Variational and optimal control problems with delayed argument

This paper deals with variational and optimal control problems with delayed argument and presents analogs of the classical necessary conditions for optimality for problems in (n + 1)-space. It is mainly concerned with the functional $$J(y) = \int_a^b {f[t,y(t\user2{--}\tau ),y(t),\dot y(t\user2{--}\tau ),\dot y(t)] dt} $$ There are no side conditions; τ is a positive real number; andy is a continuous piecewise smooth vector function havingn components. The fundamental lemma of the calculus of variations is used in deriving an analog of the Euler equations. The usual construction is utilized in obtaining analogs of the Weierstrass and Legendre conditions. Also found is a fourth necessary condition involving the least proper value associated with a boundary value problem related to the second variation. A sufficient condition is obtained by the use of a simple expansion method. The last station of the paper outlines an extension of a maximal principle obtained by Hestenes to control problems which involve delays in both the state variable and the control variable.     

Existence theorems for optimal control problems involving functional differential equations

This paper extends some existence theorems of Cesari for optimal control problems to systems whose dynamics is described by functional differential equations of finitely-retarded type. We show that the proper choice of state space is the spaceE ¹×C[–, 0], where >0 represents the time-lag of the system, and that it is necessary to choose initial conditions from a compact set inC[–, 0] as well as to employ the usual growth condition.This research was accomplished in the frame of research project AFOSR-942-65 at the University of Michigan. In particular, the author would like to thank Professor L. Cesari (University of Michigan) and Professor N. Chafee (Brown University) for many helpful remarks during the preparation of the research, which forms part of the author's doctoral dissertation written at the University of Michigan.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained.     

Transformation of equations with retarded argument to equations with the best argument

The problem of choosing the best argument in the Cauchy problem for a system of ordinary differential equations with retarded argument is studied from the viewpoint of the method of continuation of the solution with respect to a parameter. It is proved that the arc length counted along the integral curve of the problem is the best argument for the system of continuation equations to be well-posed in the best possible way. A transformation of the Cauchy problem to the best argument is obtained. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 62–68, January, 1998.