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Existence theorems are proved for weak optimal solutions of problems of optimization with distributed and boundary control. Many examples are given. Application is made of recent remarks on closure properties of linear and nonlinear operators. Recent geometric, topological, and analytical views are brought to bear on the underlying seminormality conditions.

In the theory of optimization, in connection with ordinary and partial differential equations, a number of closure and lower closure theorems have been obtained in different contexts and under a variety of conditions and modes of convergence. In particular "seminormali ty" conditions (property (Q) and its variants) have played different roles. In this paper we first prove closure and lower closure theorems for orientor fields in a rather abstract context, all based on weak convergence and MAZUR'S theorem (w167 and 5). In the context of orientor fields, these theorems can be given the most satisfactory formulation and simplest proofs (see, e.g., th. (4.i), (5.i)). Furthermore, in the present new approach, the interplay of "seminormali ty" conditions and modes of convergence can be easily seen: the stronger the mode of convergence, the weaker are the "seminormali ty" conditions that are needed. From these theorems we then derive, as corollaries, closure and lower closure statements for Mayer and Lagrange problems (~ 6 and 7) and lower semicontinuity statements for free problems (w Under suitable hypotheses, no seminormality condition (or property (Q)) is needed. Further theorems without seminormality conditions, as well as other details, are discussed in [3]. In particular, we show that seminormality conditions can be removed, not only under standard Lipschitz requirements, as expected, but also under much more satisfactory simple growth conditions, as proposed some time ago by E. H. ROTHE for free problems. Applications to multidimensional Lagrange problems are discussed in [8].

Existence theorems are proved for Lagrange problems of optimization in a given domain G with possibly unbounded distributed controls in G and on the boundary of G, and with functional relations on G and on the boundary represented by closed operators, not necessarily linear. The case where the functional relations are partial differential equations is emphasized. Recent work concerning the reduction or elimination of seminormality requirements is taken into account. Many examples are given.

This paper focuses on certain analytic criteria given by the authors in earlier works, for the geometric property of upper semicontinuity of set-valued functions, used in the proofs of lower closure theorems, and hence in existence theory. In particular, it is observed that, under Filippov-type condition (namely, when the set of controls is bounded in measure or in norm), mere Caratheodory-type continuity of the relevant functionsf is sufficient to guarantee a weak form of property (Q), and in turn the lower closure theorems.

where E: D(E) -+ Y, B(E) C X, N: X + Y, A: X+ Y are operators, E linear, not necessarily bounded, N and A continuous, not necessarily linear, X, Y Banach spaces over the reals, 01 a real parameter. In any application E may be a linear differential operator in some domain G C E*, v 3 1, with linear homogeneous boundary conditions. We consider the case in which Ex = 0 has a nontrivial set X0 = ker E of solutions; in other words, the equation Ex + hx = 0 has X = 0 as an eigenvalue. We assume, however, that X,, is finite dimensional, thus, 1 0, C > 0 such that, for every real ol with 1 ti / < tiO , the equation Ex + ol Ax = Nx has at least a solution x E X with 11 x /I < C (existence of solutions across a point of resonance). In other words, the parameter 01 is allowed to go through the point of resonance OL = 0, and yet uniformly bounded solutions x of (1) can be guaranteed. This phenomenon has physical significance. For the case of periodic solutions

ABSTRACr. Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.

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 classesL p andL ∞ .

In the line of Lyapunov’s convexity theorem, the author considers the subsets H and K obtained from any number h of given integrable n-vector functions by all possible set decompositions and by all possible convex combinations respectively and integration. Both sets are closed and convex, and the author gives a simple proof that $H = K$.

The authors prove existence theorems for the minimum of multiple integrals of the calculus of variations with constraints on the derivatives in classes of BV possibly discontinuous solutions. To this effect the integrals are written in the form proposed by Serrin. Usual convexity conditions are requested, but no growth condition. Preliminary closure and semicontinuity theorems are proved which are analogous to those previously proved by Cesari in Sobolev classes. Compactness in L1 of classes of BV functions with equibounded total variations is derived from Cafiero-Fleming theorems.

On obtient des conditions suffisantes pour l'existence de solutions doublement periodiques de l'equation d'onde non lineaire u ts =f(t,s,u), u(t+T,s)=u(t,s)=u(t,s+T) ou f est une fonction continue donnee dans R 3 et f est T-periodique en t et s

In the present paper, the author discusses an abstract formulation of control problems involving general operators ℒ :S →V, ℳ :S →Y from a Banach spaceS into spaceV andY of vector functions in a fixed domain with components inL p ,p⩾1. For this general formulation, the author states closure theorems, lower closure theorems, and existence theorems for an optimal solution. It is then shown that the problems of control involving Dieudonne-Rashevski partial differential equations previously considered by the author are particular cases of the present formulation. Finally, it is shown by examples that problems of control involving usual partial differential equations, linear or not, as well as other functional relations, can be framed in the present formulation. The present work concerns problems withdistributed controls. Work concerning problems withdistributed as well as boundary controls is forthcoming.

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.

In the last years a great deal of research has been done on the axiomatization of the concept of linear integral over a variety T, but very little for the nonlinear integrals 1=fTf(P, q), p = (Pi, * * * , PM), q = (qi, * * * , qk), though they have their natural place in Volterra's functional analysis. In the present paper we give the first elements of a very general axiomatization concerning such integrals. This is done by introducing a concept of quasi normality (quasi additivity, quiasi subadditivity, etc.) which extends Banach's concept of normality. As we will see, if 4 (I) is a quasi additive set function of bounded variation, then the integral 3(f, T, 4) of a function f(p, q) over any continuous parametric mapping T with respect to 4 (f positive homogeneous of degree one in q but not necessarily linear) can be defined by a process of limit on the set function d1(I) =f [T(r), p (I) ], rCI, and this set function is proved to be itself quasi additive [?6, Theorem (i) ]. It appears, therefore, that the concept of quasi additivity is a generalization of the usual additivity which is reproduced by the integrand f, though the latter is not linear. The integral 3 is obtained by means of a process of limit involving a mesh a which is defined axiomatically. Such a process is more general than usual since a is not required to decrease nor approach zero by refinements only. Examples of this situation are known (?4). In successive papers we will discuss the problem of extension and other properties of the integral 3 in connection with measure theory, Radon-Nikodym derivatives, and weak convergence. The present integral 3 contains, as a particular case, the concept of integral over a continuous surface T of finite area, as defined by the author and used in surface area theory and the calculus of variations by L. Sigalov, J. M. Danskin, V. E. Bononcini, J. Cecconi, L. H. Turner, and the author. In this particular situation 45(I) is the usual vector signed area of the same mapping T. As is natural, the same integral 3 contains, as a particular case, the classic Weierstrass integral over a curve of finite Jordan length. Finally, the process of limit considered in the present paper can be thought of as extending Burkill's integration process for normal interval functions. Burkill's integral has been used in connection with total variation, Jordan

Introduction. In the present paper II we prove existence theorems for weak optimal solutions of nonparametric Lagrange problems with (or without) unilateral constraints. In paper I ([le] of the reference list) we considered arbitrary pairs x(t), u(t) of vector functions, u(t) measurable with values in Em, x(t) absolutely continuous with values in En, and we discussed the existence of the absolute minimum of the functional

Introduction. In the preceding paper [1 ] we have introduced by means of axioms a concept of quasi additive vector-valued set functions +(I) = (41, * * *, Ok) in a class { I} of sets I of a space A, a concept of mesh (D) of certain finite collections D of sets I, and we have shown that an integral a(f, T, 4) of a function f(p, q) over a variety T: p = p(w), w A, with respect to the quasi additive set function +(I), can be obtained by a standard process of limit over quasi additive set functions as 5(D)->0. Here f(p, q), pEE., qCEk, is a function with f(p, tq) = tf(p, q) for all t > 0, satisfying usual hypotheses of continuity. In the present paper, we discuss the problem of extension of quasi addi

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 constantMźź0 is replaced by any functionMź(t)ź0 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 ź|xź|pdtźM, which are required to be satisfied by the admissible pairs of the class Ω.

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)

In previous papers [3-5] we have given existence theorems for solutions in Sobolev spaces of multidimensional Lagrange problems of optimization in a fixed domain, bounded or unbounded. In the present paper we shall apply the previous results to the case where the partial differential equations are written in canonic form. The canonic form which we take into consideration is the one proposed by RASHEVSKY in his book [9, pp. 323 324]. THOMAS & TITT [10] consider essentially the same form though this form is not written explicitly. ILaSHEVSKY has pointed out the generality of the proposed form. In [6-8] LURm discusses necessary conditions for Lagrange problems of optimization with partial differential equations in canonic form. In particular, in [7, 8] LumE discusses necessary conditions for an optimal solution in a two dimensional problem of magnetohydrodynamic channel flow, though with no previous existence analysis. In the present paper we shall show that our existence theorems yield the existence of an optimal weak solution for LURIE'S problem of magnetohydrodynamics. This optimal solution, of which we prove the existence, is weaker, however, than those considered by LURIE, namely it is a weak solution in the sense of Gamkrelidze with state functions belonging to a Sobolev space W21.

(1958). Rectifiable Curves and the Weierstrass Integral. The American Mathematical Monthly: Vol. 65, No. 7, pp. 485-500.

This paper concerns the concept of upper semicontinuity of variable sets, precisely the variant of Kuratowski's 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 Tonelli's 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.

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~

where E is a real elliptic linear differential operator in a bounded domain G of R” with a given system of linear homogeneous conditions, say, BX = 0 on the boundary aG of G and where N is a Nemitsky type nonnecessarily linear operator. We shall make use here of the alternative method, and particularly we shall make use for the elliptic case of new remarks. These remarks suggest that both the auxiliary and bifurcation equations can be analyzed under different topologies, and by a more specific construction of the operator S: Y0 -+ X0. Actually some of these remarks have been already used implicitly in previous papers on the semilinear wave equation in R2 (Cesari and Kannan [5], Cesari and Pucci [6]). For selfadjoint elliptic problems, Landesman and Lazer [9] proved, also by the alternative method, a remarkable theorem which was then extended by Williams [14] by the same method, and by others by different arguments. Later, Shaw [12] proved, again by the alternative method, that Landesman’s and Lazer’s theorem extends even to nonselfadjoint problems with equal Fredholm indices and whose eigenfunctions share regions of positivity and negativity with their corresponding adjoint eigenfunctions. In the present paper we definitely aim at elliptic problems which are not necessarily selfadjoint and do not necessarily satisfy Shaw’s requirements. The sufficient conditions we obtain are more quantitative in character and concern the cases Nx = f(t) + g(t, D”x) and NX = f(f) + g(t, x(t)), t E G. However, as we show by examples for the case Nx = f(t) + g(t, x(t)), our sufficient conditions for existence allow a great freedom on g, on which no monotonicity is required.

In this paper we apply the direct method of the calculus of variations, based on lower semicontinuity and lower closure (see [5]), to prove the existence of optimal solutions x(t) = (x 1,..., x n ), t 1 ≤ t ≤ t 2, for which a components y(t) = (x 1,..., x α ) are AC and n — α components z(t) = (x α+ 1,..., x n ) are BV and not necessarily AC. If α = 0 all components of x are BV, and in this situation no growth assumption is made on the integrand function. The cost functional P is of Serrin type ([14]), i.e. it is obtained from the usual integral expression I by means of a limit process, based on a topology τ, of this integral I over curves x k whose components are all AC. The topology τ that we use here is the topology of uniform convergence on the y k components and pointwise convergence almost everywhere on the z k components. This pointwise convergence almost everywhere has been used by Cesari in the study of area of discontinuous surfaces ([4], 1936) and in existence theorems concerning simple integrals for AC solutions (see [5], Chapt. 15 and the papers cited there).

On considere le probleme U tt -U ττ =Φ(t,τ)+g(u), (t,τ)∈R 2 u(t+T,τ)=u(t,τ)=u(t,τ+T), Φ(t+T,τ)=Φ(t,τ)=Φ(t,τ+T) ou u et Φ sont des fonctions scalaires de R 2 , et g une fonction scalaire sur R. On suppose que g et Φ sont continues et on determine des conditions suffisantes sous forme d'inegalites pour garantir que ce probleme a des solutions continues

The authors give an elementary proof of an equivalence theorem of analysis which is often used in optimization theory. The theorem asserts that certain conditions are equivalent to weak convergence inL1. One is the Dunford-Pettis condition concerning absolute integrability. Two others are expressed in terms of Nagumo functions, and can be thought of as growth properties. The original proofs of the various parts of the theorem are scattered in different and specialized mathematical publications. The authors feel it useful to present here a straightforward proof of the various parts in terms of standard Lebesgue integration theory.