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Let A be a closed subset of the tx-space E1 × En, t∈E1, x = (x1,…xn)∈En and for each (t, x)∈A, let U(t,x) be a closed subset of the u-space Em, u = (u1,…,um). We do not exclude that A coincides with the whole tx-space and that U coincides with the whole u-space. Let M denote the set of all (t, x, u) with (t, x)∈A. u∈U(t, x). Let f(t, x, u) = (f0, f) = (f0, f1,…, fn) be a continuous vector function from M into En+1. Let Bbe a closed subset of points (t1x1,t2,x2)of E2n+2, x1 = (x1 1,…x1 n, x2 = (x2 1,…x2 n.We shall consider the class of all pairs x(t), u(t), t1 ≤t≤t2, of vector functions x(t), u(t) satisfying the following conditions : (a) x(t) is absolutely continuous (AC) in [t1, t2]; (b) u(t) is measurable in [t1, t2]; (c) (t,x(t))∈A for every t∈[t1, t2]; (d) u(t)∈U(t, x(t)) almost everywhere (a.e.) in [t1, t2]; (e) f0 (t,x(t), u(t)) is L-integrable in[t1, t2]; (f) dx/dt / f(t, x(t), u(t)) a.ein [t1, t2]; (g) (t 1,x(t 1), t 2, x(t 2))∈B.

I(x)

We continue here the discussion on the existence of discontinuousBV minima, for a class of multiple integrals of the calculus of variationsI, we have started in [2] in view of possible studies on hyperbolic partial differential equations. Besides the associated Serrin integralJ, based onL1-convergence, we take into consideration a modified Serrin-type functionalJ*. This new integralJ* will be needed in [3] to prove Rankine-Hugoniot type properties.

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.

By properties of operators of the Brezis type (m), monotone operator theory, and fixed point theorems, the authors prove existence theorems for solutions, and existence theorems for optimal solutions, in problems monitored by nonlinear abstract parabolic type equations. Applications are made to Cauchy-Dirichlet and to Cauchy-Neumann problems, in a cylinder. It is shown that seminormality requirements of previous papers can be dropped. In an alternative approach, controllability results are obtained by a direct application of Kakutani type fixed point theorems.

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].

We prove here a Filippov type implicit function theorem for Carath$eacute;odory mappings in general topological space. We also prove a Banach space version which is applicable to optimal control problems. For this purpose, we introduce the basic concepts of separation in topological spaces. Connections between separation properties of spaces and measurability of graphs of multifunctions are established. AMS (MOS): 57C65

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.

A known existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is being proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.

Si prendono in considerazione sistemi iperbolici di equazioni differenziali del primo ordine alle derivate parziali nelle forme canoniche di Courant-Lax e di Schauder (nn. 1 e 2). Per tali sistemi in uno strato sottileDa=[0≤x≤a, y ∈Er] dello spazioEr+1 si formulano certi problemi ai limiti e si enunciano corrispondenti teoremiA eB di esistenza, unicita, e dipendenza continua dai dati. In particolare, se i dati sono tutti periodici nelle coordinatey1,...,yr, allora anche le soluzioni di cui si dimostra l'esistenza sono periodiche e degli stessi periodi. I teoremiA eB contengono ipotesi specfiche che per semplicita possono essere espresse qui dicendo che le matrici in considerazione debbono avere «diagonale principale dominante». Nel no. 3 si mostra mediante esempi che le conclusioni dei teoremi A e B possono cadere in difetto se queste condizioni non sono soddisfatte. Nel no. 5 si prende in considerazione un problema ai limiti del tipo sopra considerato che e stato proposto come semplice modello preliminare per un fenomeno assai complesso di risonanza in ottica non lineare. Mediante trasformazioni algebriche si mostra nel no. 5 che il problema puo essere ridotto ad uno del tipo considerato nel no. 2 con matrici a diagonale principale dominante per cui vale il teorema B di esistenza e di unicita.

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.

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.

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 present paper the authors present a number of results concerning closure, lower closure and lower semicontinuity in problems of optimal control. In these theorems the convexity of the relevant sets plays a major role, together with certain further conditions, usually denoted as “semi–normality” conditions, among which are, in particular, properties of upper semicontinuity of variable sets. Particular emphasis is given to recent results where seminormality conditions have been reduced to a minimum, or not used at all. We also show that certain analytical conditions of practical significance, recently used in lieu of seminormality, imply a weak form of seminormality.

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 a number of existence theorems for Lagrange problems with unbounded controls certain growth conditions are required, which generally can be omitted for bounded controls. In the present paper the authors show that even for unbounded controls the growth conditions can be dispensed with in an arbitrary subset in the $tx$-space $E_{1 + n} $, provided such a subset is “slender” according to suitable definitions. Any set contained in finitely many or countably many smooth curves $x = \phi (t)$, $a \leqq t \leqq b$, is certainly slender. Many examples are given.

Si dimostra che ogni trasformazione piana, rappresentata da funzioni assolutamente continue secondoTonelli con dervate parziali prime integrabiliL 2→ α con α>0, sono a variazione limitata e assolutamente continue secondoBanach eVitali. Si danno inoltre esempi effettivi di trasformazioni piane, rappresentate da funzioni assolutamente continue secondoTonelli con derivate parziali prime integrabiliL 2, che non sono ne a variazione limitata ne assolutamente continue secondoBanacheVitali.

THIS PAPER attempts to focus on a number of areas in nonlinear differential equations which appear to be of heightened interest today.

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.