Lamberto Cesari

Existence in the large of periodic solutions of hyperbolic partial differential equations

where u = (u 1 . . . . , u.), and f = (fl . . . . . f . ) is periodic in x and y of period T, presents a number of difficulties when no damping of any sort is assumed. In this paper we analyze this difficult problem in the line of our previous work on ordinary and partial differential equations. We conclude with criteria of existence for solutions to the problem above. These criteria can then be used for the analogous problem for the equation uxx-u,y=g(x, y, u, ux, u~,). (2)

Optimization with partial differential equations in dieudonné-rashevsky form and conjugate problems

usual boundary conditions, and constraints u~U(t, z)~Em, where the control space U = U(t, z) is a given subset of Era. Here z =(z 1 . . . . . : ) are state variables and u =(u 1 .. . . , u m) control variables. As stated by P. K. RASHEVSKY [6], very general partial differential equations and systems can be written locally in the form (2). For instance, second order partial differential equations such as z~,+z~s= 1 u) can be immediately written in the form f ( t , s, z 1, zlt, z~,

ALTERNATIVE METHODS IN NONLINEAR ANALYSIS

Publisher Summary This chapter discusses a few alternative methods in nonlinear analysis. Alternative methods, based on functional analysis, have recently undergone an intensive development through the work of many researchers. The bifurcation theory is being used in theoretical existence analysis of the solutions, in methods of successive approximations of the solutions, and in estimating the error of approximate solutions. In terms of functional analysis alone, the equation is being decomposed into a system of two equations, one of which, usually called the auxiliary equation, is in a ∞-dimensional space and is often uniquely solvable. By substitution, one is then reduced to the solution of the other equation called the bifurcation or determining equation, which possibly lies in a finite-dimensional space.

Periodic Solutions of Hyperbolic Partial Differential Equations

Publisher Summary This chapter discusses the periodic solutions of hyperbolic partial differential equations. In the chapter, the classes of hyperbolic partial differential equations with fixed or variable fixed characteristics are taken into considerations. The existence theorem for hyperbolic partial differential equations is verified and proved on the basis of Schauder fixed point theorem. The proof is more difficult than the corresponding proof for the Darboux problem as the requirements for m make the usual estimates inapplicable.

An Existence Theorem without Convexity Conditions

We state and prove existence theorems for problems of optimal control which are linear in the state variables. As in previous work by L. W. Neustadt and C. Olech, no convexity condition is required. Examples are given.

Seminormality and upper semicontinuity in optimal control

This paper concerns the concept of upper semicontinuity of variable sets, precisely the variant of Kuratowskis definition of upper semicontinuity that Cesari has denoted as property (Q). This concept has been used by Cesari in most of his papers on existence theorems for optimal solutions, and later used by Olech, Lasota and Olech, Brunovsky, Baum, Suryanarayana, and Angell. First, criteria are given for property (Q) in addition to those which had been already given previously. Then, it is shown that a slight restriction in the concept can be expressed in a form which is similar to Tonellis concept of seminormality for free problems of the calculus of variations. Thus, the property (Q) appears to be a generalization to Lagrange problems of control of the well-known concept of seminormality for free problems.

Closure theorems without seminormality conditions

The authors give a variety of conditions under which there is no need to explicity require seminormality conditions in closure and lower closure theorems, and corresponding lower semicontinuity theorems. Both Lipschitz-type conditions and growth-type conditions are taken into consideration in classesLp andL∞.

Closure, Lower Closure, and Semicontinuity Theorems in Optimal Control

First we discuss in detail the concept of lower closure for Lagrange problems, and we extend in various ways previous closure and lower closure theorems. In particular, we show that the present lower closure theorems and related concepts are extensions to Lagrange problems of optimal control of well-known semicontinuity theorems for free problems of the calculus of variations and the related concept of seminormality. Finally, we prove by a new approach that the convexity condition usually requested in lower closure theorems is, in a suitable sense, both a necessary and sufficient condition for lower closure.

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.

Remarks on some existence theorems for optimal control

First, a remark is made that a growth condition contained in previous papers by Cesari concerning existence theorems for optimal controls can be replaced by a slightly more general condition. In this more general condition, a constantMɛ⩾0 is replaced by any functionMɛ(t)⩾0 which is assumed to beL-integrable in every finite interval.Then, the remark is made that the same condition, which is usually required to be satisfied by the functionsf0(t, x, u),f(t, x, u) characterizing the control, can be required to be satisfied only by the admissible pairsx(t),u(t) of the class Ω in which the optimum is being sought. This generalization requires a subtle argument. The new condition parallels now the usual conditions of the type ∝|x′|pdt⩽M, which are required to be satisfied by the admissible pairs of the class Ω.