Optimal control and relaxation of nonlinear elliptic systems

In this paper we study the optimal control of systems driven by nonlinear elliptic partial differential equations. First, with the aid of an appropriate convexity hypothesis we establish the existence of optimal admissible pairs. Then we drop the convexity hypothesis and we pass to the larger relaxed system. First we consider a relaxed system based on the Gamkrelidze-Warga approach, in which the controls are transition probabilities. We show that this relaxed problem has always had a solution and the value of the problem is that of the original one. We also introduce two alternative formulations of the relaxed problem (one of them control free), which we show that they are both equivalent to the first one. Then we compare those relaxed problems, with that of Buttazzo which is based on the -regularization of the extended cost functional. Finally, using a powerful multiplier rule of Ioffe-Tichomirov, we derive necessary conditions for optimality in systems with inequality state constraints.Research supported by NSF Grant DMS-8802688     

Discrete approximation of relaxed optimal control problems

We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Relaxation and existence of optimal controls for systems governed by evolution inclusions in separable Banach spaces

In this paper, we examine relaxed control systems governed by evolution inclusions in a separable Banach space. First, we establish the existence of admissible trajectories, correcting an earlier result of Ahmed. Then, we obtain a compactness result for the set of admissible trajectories. Using this compactness result, we prove the existence of optimal solutions for optimal control problems; furthermore, we show that the values of the original and relaxed problems are equal. Finally, we show that the original trajectories are dense in the set of relaxed trajectories. An example is worked out.This research was supported by NSF Grant No. DMS-86-02313.     

Optimal control,sensitivity analysis and relaxation of maximal monotone integrodifferential inclusions inR N

In this paper we examine optimal control problems governed by maximal monotone integrodifferential inclusions inR ^N . First we establish the existence of an optimal control. Then we show that the value of the problem depends continuously on a parameter appearing in all the data. Then we introduce the relaxed system, we show that under very general hypotheses it has a solution and that its value equals that of the original problem. Subsequently we show that relaxability and performance stability are equivalent concepts. Finally we specialize our results to the class of controlled differential variational inequalities.Research supported by NSF Grant DMS-8802688     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

On a unifying approach to decomposition theorems of Yosida-Hewitt type

In this paper we deal with a very general form of the Yosida-Hewitt theorem on the decomposition of measures into countably additive («normal») and purely finitely additive («antinormal») parts. It expands a previous one by the authors with the aim of joining two different standpoints to the Yosida-Hewitt type theorems. The first goes back to the original publication defining the «antinormal» part as a certain disjoint complement to the «normal» one. The second approach goes deeper and characterizes this disjoint complement intrinsically i.e. as a measure, functional or operator which is equal to zero on a huge set. These two points of view are common for the publications connected, respectively, with measure theory and, theory of vector lattices; the second allows important applications. The unification of these approaches gives an opportunity to derive new information in the case of vector measures. We have taken the opportunity of this paper also to furnish a survey of the topic.     

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

Summary The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature ^e. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R,     

Sui probleimi ellittici non lineari con coefficienti crescenti «rapidamente» o «lentamente»

Summary We study boundary value problems in variational form for nonlinear elliptic operators in the case the nonlinearity is not of polynomial type. We give some examples of applications of abstract theorems for homogeneous problems obtained by other authors (e.g. Donaldson, Grossez) to homogeneous «intermediate» and «mixed» problems. Furthermore we prove some existence theorems for non homogeneous problems.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Anatomy of the Shape Hessian

Summary The computation of the Shape Gradient with respect to domain perturbations plays a central role in the theory and numerical solution of Shape Optimization problems. In 1907 J. Hadamard introduced a method which has been and still is widely used to obtain many useful results for applications. The mathematical limitation of his method rests in the fact that the deformations of the domain are a function of the smoothness of the normal to the boundary (hence the smoothness of the boundary). New developments by the Nice School (J. Cea and J. P. Zolesio) using arbitrary velocity fields of deformation relaxed the condition that the deformation be carried by the normal to the boundary. Finally the use of «Shape Lagrangians» by Delfour and Zolésio made it possible to obtain Shape Gradients by a simple constructive method which does not require the derivative of the state with respect to the domain. In this paper we apply this last method to semi convex cost functions. This extension makes it possible to compute the «Shape Hessian» or «Shape directional second derivative». We give several expressions for the «Shape Hessian» and a set of equations characterizing its kernel.

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Résumé En Optimisation de forme le calcul du gradient par rapport aux déformations du domaine joue un rôle central dans la théorie et la résolution numérique de ces problèmes. C'est en 1907 que J. Hadamard montra comment obtenir ces gradients par une méthode originale de perturbations du domaine portées par la normale à la frontière du domaine. La «méthode d'Hadamard» a permis jusqu'à nos jours de traiter ces problèmes et d'obtenir de nombreux résultats utiles pour les applications. La limitation mathématique de cette méthode réside dans le fait que les déformations sont liées à la régularité de la normale (donc celle de la frontière). De nouveaux développements utilisant la méthode des champs de vitesse de déformation de l'École niçoise (J. Céa et J. P. Zolésio) ont permis de séparer les contributions respectives de ces deux variables. Enfin l'utilisation du «lagrangien de forme» par Delfour et Zolésio a permis d'obtenir les gradients de forme par une méthode simple et constructive ne nécessitant pas la dérivée de l'état par rapport au domaine. Dans cet article nous appliquons cette dernière méthode à des fonctions coût semi convexes ce qui permet d'obtenir l'«hessien de forme» ou derivée directionnelle seconde par rapport aux domaines. On donne plusieurs expressions de cet hessien ainsi que des équations caractérisant son noyau.

     

Wiener criterion and potential estimates for obstacle problems relative to degenerate elliptic operators

Summary We give a Wiener's type criterion for the continuity of the local solutions of obstacle problems relative to a degenerate elliptic operator. Moreover, we give an estimate on the modulus of continuity of the solutions and we also estimate the «energy decay» at a point.

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Sunto Si dà un criterio di Wiener per la continuità in un punto delle soluzioni locali di problemi d'ostacolo relativi ad un operatore ellittico degenere. Si ottiene inoltre una stima del modulo di continuità della soluzione e del «decay» dell'«energia» in un punto.

     

Optimal control and admissible relaxation for subdifferential evolution inclusions

In this paper we study the relaxation of optimal control problems monitored by subdifferential evolution inclusions. First under appropriate convexity conditions, we establish an existence result. Then we introduce the relaxed problem and show that it always has a solution under fairly general hypotheses on the data. Subsequently we examine when the relaxation is admissible. So we show that every relaxed trajectory can be approximated by extremal original ones (i.e. original trajectories generated by bang-bang controls) and that the values of the original and relaxed problems are equal. Some examples are also presented.     

On infinite-dimensional control systems with state and control constraints

In this paper we examine infinite-dimensional control systems governed by semilinear evolution equations and having both state and control constraint. We introduce the relaxed system and show that the original trajectories are dense in an appropriate function space in the relaxed ones. We also determine the dependence of the solution set on the initial conditions. Then using those results we establish necessary and sufficient conditions for optimality for some optimization problems. Finally we prove some controllability results.     

Una struttura sintattica unitaria per le teorie della relatività ristretta e della meccanica classica

A unified axiomatic scheme for both the Newtonian Mechanics and the Special Theory of Relativity is given, by setting two systems of Axioms that differ from each other in only one requirement about the possibility of measuring time-intervals by light reflections. The concept of «observer» is obtained as a derived concept, rather than a primitive one, as in some previous papers by other Authors. The status of Newtonian Mechanics as a «limiting case» of Special Relativity is rigorously deduced as a consequence of the result that the geometric structure of (neo)classical space-time is a limit of a family of relativistic geometric structures for the space-time.     

On necessary optimality conditions for systems Governed by a two point boundary value problem. I

The paper concerns a necessary optimality condition in form of a Pontryagin Minimum Principle for a system governed by a linear two point boundary value problem with homogeneous Dibichlet conditions, whereby the control vector occurs in all coefficients of the differential equation. Without any convexity assumption the optimality condition is derived using a needle-like variation of the optimal control. In case of convex local control constraints the optimality condition implies the linearized minimum principle, which we have proved in [2]. An example shows that for this linearized optimality condition the convexity of the set of all admissible controls is essential.     

OnBMO regularity for linear elliptic systems

Summary We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO()», a class neither containing, nor contained in . We also prove local and global L^p regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.Work partially supported by M.P.I.Project 40% «Equazioni di evoluzione e applicazioni fisico-matematiche».     

On partial asymptotic stability by the method of limiting equation

Summary The extensions of the Barbashin-Krasovskij theorem to the partial asymptotic stability of the zero solution of a differential system require the boundedness of the uncontrolled coordinates along the solutions. In this paper the Barbashin-Krasovskij method is generalized without supposing «a priori» knowledges on the solutions. At the same time, the results extend one of C. Risito's theorem to nonautonomous differential equations. As an application, stability properties of the equilibrium state of nonholonomic dissipative mechanical systems are studied.     

Two theorems in classical vorticity theory

Zusammenfassung Es wurden zwei neue Sätze für den Transport von Wirbeln aufgestellt. Dabei zeigt sich, dass die Begriffe «Konvektion» und «Diffusion» neu definiert werden sollten.     

Asymptotic expansions obtained by a center manifold theorem

Sunto Viene presentato un nuovo metodo per la determinazione degli sviluppi asintotici della soluzione esterna di sistemi di equazioni differenziali ordinarie singolarmente perturbati. Il metodo proposto, basato sulla teoria geometrica delle perturbazioni singolari e in particolare su un teorema di esistenza di varietà centrale, permette di ottenere le equazioni differenziali che definiscono le variabili « lente » senza la preventiva conoscenza dei corrispondenti sviluppi per le variabili « veloci ». Inoltre, se i sistemi vengono dati con condizioni iniziali, alcune formule che esprimono le corrette condizioni iniziali da assegnare alle equazioni differenziali trovate — formule già note nel « caso stabile » — vengono estese al « caso condizionalmente stabile »; il procedimento qui usato risulta anche più sintetico rispetto a quelli precedentemente proposti. Infine viene studiata un'applicazione ad una classe assai generale di equazioni derivanti dalla cinetica delle reazioni enzimatiche.

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Lavoro eseguito nell'ambito dei programmi del gruppo di ricerca « Equazioni di Evoluzione e Applicazioni », M.P.I., e del Gruppo Nazionale Fisica-Matematica del C.N.R.