Convexity Hypothesis Research Articles

On the optimal control and relaxation of nonlinear infinite dimensional systems

In this paper we study optimal control problems for infinite dimensional systems governed by a semilinear evolution equation. First under appropriate convexity and growth conditions, we establish the existence of optimal pairs. Then we drop the convexity hypothesis and we pass to a larger system known as the « relaxed system ». We show that this system has a solution and the value of the relaxed optimization problem is equal to the value of the original one. Next we restrict our attention to linear systems and establish two « bang-bang » type theorems. Finally we present some examples from systems governed by partial differential equations.

Relaxation and Optimal Control of Nonlinear Distributed Parameter Systems

In this paper we examine optimal control problems governed by nonlinear evolution equations. First we establish the existence of optimal controls under a convexity hypothesis on an appropriate orientor field. Then we drop the convexity hypothesis. Now in order to guarantee the existence of optimal solutions we need to pass to a larger system known as the "relaxed system". We introduce that system, show that it has optimal solutions and compare it with original one. Finally using the Dubovitsky-Milyutin formalism we obtain necessary optimality conditions (maximum principle) and we prove a new "bang-bang" theorem.

Relaxation of infinite dimensional control systems

IT IS well known that convexity is crucial in the solution of optimal control problems. Namely, if the orientor field determined from the deparametrization operation is convex valued and satisfies some kind of set valued upper semicontinuity hypothesis, then one can prove the existence of optimal controls. The first such existence result was proved by Filippov [ 141 for a time-optimal control problem in IR”, with a control vector field that is continuously differentiable in the state variable [ 14, theorem 11. In that same paper, Filippov presents an example of a time optimal control problem that does not satisfy the convexity hypothesis and so does not have an optimal solution (see Section V of [14]). What happens in that example, is that the minimizing sequence of optimal controls (u,( -)I, z , , rapidly oscillates between + 1 and 1, remaining at each value for the same length of time. Then the Dirac measures 6u,(-), converge weakly (narrowly in the Bourbaki terminology), to the probability distribution &I, + $3_, . This suggests that we have to consider an extension of the original variational problem with measured valued controls, which satisfies the convexity requirement. This extension of control problems, was introduced independently by Gamkrelidze [15], Ghouila-Houri [ 171 and Warga [27]. Gamkrelidze called the controls of the extended problem “sliding regimes” or “chattering controls”, while Warga called them “relaxed controls”. Here we follow the terminology of Warga, which appears to be more widely used, at least in the western literature. We should also mention that other existence results like theorem 1 of Filippov [14] were also proved by Roxin [24] and Wazewski [29]. Later Berkovitz, Cesari, Olech and others improved those results. A very lucid account of the existing theory with complete bibliographical information, can be found in the books of Berkovitz [2] (in particular Chapters III and IV) and Cesari [12]. A detailed study of the relaxed problem, can also be found in the books of Gamkrelidze [ 161 and Warga [28], In the late sixties, Cesari in a series of important papers (see [8-111 and the references therein), addressed the question of existence of optimal controls for distributed parameter systems. Throughout his existence results, Cesari had a convexity hypothesis in the form of the well known by now, property (Q). Our work in this paper can be viewed to a certain extend, as a continuation of the work of Cesari 18-l 11. Namely we examine what happens when the con_ _ vexity hypothesis is no longer satisfied. If this is the case, then we can always convexify the orientor field and this in a control theoretic level corresponds to the introduction of time

A stability theorem for entropy solutions of initial value problems for first order quasilinear hyperbolic systems in several space variables

In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u 1 (x, t),..., u r (x, t)),f i =(a 1 i (x, t, u),..., a r i (x, t, u)), i=1,...,n, g=(g 1 (x, t, u),...,g r (x, t, u),i G ℝ n and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf u i are symmetric and the Hessian(a j i ) uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL loc 2 within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (−,x i ,−,t), (−,y i ,−,t) on a compact setD ⊂ ℝ n x (0, ∞) and a functionK(t)∈L loc 1 ([0, ∞)) depending onD. Here we denote by (−,x i ,−,t) and (−,y i ,−,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ℝ n x [0,T].

Théorèmes d'existence en calcul des variations et applications à l'élasticité non linéaire

SynopsisIn this paper, we study problems of the form:We obtain some existence and regularity results when Ω is either a ball or an annulus, without convexity hypothesis on g. We then apply these results to some shear problems in nonlinear elasticity.

A general Farkas lemma and characterization of optimality for a nonsmooth program involving convex processes

In this paper, a generalization of the Farkas lemma is presented for nonlinear mappings which involve a convex process and a generalized convex function. Using this result, a complete characterization of optimality is obtained for the following nonsmooth programming problem: minimizef(x), subject to − ∈H(x) wheref is a locally Lipschitz function satisfying a generalized convexity hypothesis andH is a closed convex process.

Usefulness of Pretests for Estimating Underlying Technologies Using Dual Profit Functions

Translog and generalized-linear approximating forms are used to estimate dual profit functions from stochastically simulated data generated from a known, relatively complex technology under profit-maximizing conditions. The hypotheses of monotonicity, convexity, and equality of parameters common to the profit function and demand equations are tested. This last hypothesis is always rejected and monotonicity is never rejected. Convexity is rejected for about a third of the translog models and fewer than 5 percent of generalized-linear models. Neither approximating form strongly dominates the other for accuracy in estimating the underlying elasticities of substitution. Copyright 1987 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Optimal Foraging in Nonpatchy Habitats. 2: Unbounded One-Dimensional Habitat

We want to determine the trajectory that an animal must follow in order to maximize its food intake. In this paper, the habitat is supposed to be one-dimensional and infinite. The food distribution on this habitat can be arbitrary (continuous or not). The animal has a limited time T available to exploit the food resource and to return to its starting point. We find explicitly the optimal strategy, i.e., the stopping point $\bar x$ and the velocity at each point of the traversed segment $[ 0,\bar x ]$. This segment is divided into two subsets according to the food density. In the richer subset the animal equalizes the density distribution; in the poorer one, it travels as fast as possible.Mathematically, we approximate the food distribution by a piecewise constant distribution, and we solve explicitly the approximate problem by using the techniques of the calculus of variations based on convexity hypotheses. Using then a density argument we recover the solution of the general problem.

On the existence of sporadically catching-up optimal solutions for infinite-horizon optimal control problems

In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catching-up optimal solutions for autonomous, infinite-horizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the above-mentioned authors do not.

Sulla rilassazione di alcuni funzionali integrali in dimensione infinita

In this work we consider the problem (ℑ)of the minimum for integral functionals I f of the Calculus of Variations, where we don't suppose any hypothesis of convexity on the integrand and in which the state takes values in a Suslin metric space, while the last variable of the integrand is in a separable reflexive Banach space. We study two different types of relaxed problems, associated to (ℑ),and we prove that the infima of the three functionals considered are the same, that (ℑ′),one of the relaxed problems, has solution and that the accumulation points of the minimizing sequences for (ℑ)are the only solutions of (ℑ′).

A note on nondifferentiable symmetric duality

Under suitable hypotheses on the function f, the two constrained minimization problems:are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to:in which T and S are convex cones, S* is the dual cone of S, and ∂y denotes the subdifferential with respect to y. The usual method of proof uses second derivatives, which are no longer available. Therefore a different method is used, where a nonsmooth problem is approximated by a sequence of smooth problems. This duality result confirms a conjecture by Chandra, which had previously been proved only in special cases.

Nondifferentiable optimization by smooth approximations

Necessary Lagrangian conditions are obtained for nonsmooth optimization problems, involving locally Lipschitz functions and arbitrary convex cones.The approach is by a sequence of smooth local approximations to the problem, which leads to Clarke generalized gradients in the limit.The method is applied to finite dimensional problems with convex cones, semiinfinite programming problems, continuous programming problems, and optimal control, giving Pontbyaoin type conditions.A duality theorem is also obtained, with the usual convexity hypotheses weakened to a generalized invex condition, using generalized gradients.

Théorémes d'existence en optimisation non convexe

Some existence results in non convex optimization. In this article we give a method to solve, under suitable hy-pothesis , the class of problems of calculus of variations: where V i s a Sobolev space and ⋀ a differential operator . We suppose no hypothesis of convexity on g .

Perturbed minimization, with constraints adjoined or deleted

A perturbed constrained nonlinear minimization problem is considered, in spaces of arbitrary dimensions, without convexity or compactness hypotheses. Under some assumptions about the perturbed minimum, the Frechet derivative of the perturbed optimal objective function is related to the Lagrangian for the problem. This result allows bounds to be calculated for the effect of deleting some constraints. This applies, in particular, to the truncation of a semi-infinite programming problem to a finite dimensional, and thus computable, problem. Bounds for adjoining constraints, or deleting variables, are also obtained, under further assumptions.

Sufficient conditions for the optimal control of a class systems with continuous lags

Sufficient conditions in the form of a maximum principle are obtained for the optimal control of a system described by integro-differential equations and subject to some specified path constraints. The conditions are relaxed to allow for jumps in the adjoint variables at the junction points, provided a certain convexity hypothesis is satisfied for the constraint set at these points.

On disequilibrium adjustment processes

Here, we study a class of disequilibrium processes defined by differential inequalities, where prices are adjusted according to the agent's short-term demands, in both pure exchange and monetary settings. We establish convergence theorem and accessibility theorem without convexity hypotheses on the utility functions.

Sufficient Fritz John optimality conditions for nondifferentiable convex programming

AbstractThe Fritz John necessary conditions for optimality of a differentiable nonlinear programming problem have been shown, given additional convexity hypotheses, to be also sufficient (by Gulati, Craven, and others). This sufficiency theorem is now extended to minimization (suitably defined) of a function taking values in a partially ordered space, and to (convex) objective and constraint functions which are not always differentiable. The results are expressed in terms of subgradients.

A New Approach to Lagrange Multipliers

We consider a mathematical programming problem on a Banach space, and we derive necessary conditions for optimality in Lagrange multiplier form. We prove further that “most mathematical programming problems are normal.” The novelty of our approach lies on the one hand in the absence of both differentiability and convexity hypotheses on the functions delimiting the problem, and on the other hand in the method of proof, which is new. The approach unifies the well-known smooth and convex cases besides treating a new general class of problems. The main analytical tool in this paper is an extension to infinite-dimensional spaces of the “generalized gradient” previously introduced by the author. The calculus of the generalized gradient is explored as a preliminary step.

Weak solutions of nonlinear hyperbolic conservation laws

A weak solution of the conservation law ut+j(ux)=0 is a locally Lipschitzian function u which satisfies the equation almost everywhere. We treat a boundary value problem and also a mixed initial-boundary value problem associated with the equation where the initial and boundary data are convex functions. The convexity hypothesis makes it possible to apply the Fenchel theory of conjugate convex functions to the problem. This leads to a construction of solutions rather than to a proof of their existence; the solutions so constructed turn out to be stable.

Global analysis and economics III: Pareto Optima and price equilibria

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let ʌ be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between ʌ and the set of Pareto Optima.