Reconstruction of boundary controls in parabolic systems

In the paper, an inverse dynamic problem is considered. It consists in reconstructing a priori unknown boundary controls in dynamical systems described by boundary value problems for partial differential equations of parabolic type. The source information for solving the inverse problem is the results of approximate measurements of the states of the observed system’s motion. The problem is solved in the static case; i.e., to solve it, we use all the measurement data accumulated during some specified observation interval. The problem under consideration is ill-posed. To solve it, we propose the Tikhonov method with a stabilizer containing the sum of the mean-square norm and total time variation of the control. The use of such nondifferentiable stabilizer allows us to obtain more precise results than the approximation of the desired control in the Lebesgue spaces. In particular, this method provides the pointwise and piecewise uniform convergences of regularized approximations and makes possible the numerical reconstruction of the subtle structure of the desired control. In the paper, the subgradient projection method for obtaining a minimizing sequence for the Tikhonov functional is described and substantiated. Also, we demonstrate the two-stage finitedimensional approximation of the problem and present the results of numerical simulation.     

Tracking and controlling unstable steady states of dynamical systems

An adaptive controller for stabilization of unknown unstable steady states (spirals, nodes and saddles) of nonlinear dynamical systems is considered and its robustness under the changes of the location of the fixed point in the phase space is demonstrated. An analog electronic controller, based on a low-pass filter technique, is described. It can be easily switched between a stable and an unstable mode of operation for stabilizing either spirals/nodes or saddles, respectively. Numerical and experimental results for two autonomous systems, the damped Duffing–Holmes oscillator and the chaotic Lorenz system, are presented.     

On an optimal control problem involving first order hyperbolic systems with boundary controls

In this paper, we consider an optimal control problem involving a class of first order hyperbolic systems with boundary controls. A computational algorithm which generates minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated. Moreover, a necessary and sufficient condition for optimality is derived and a result on the existence of optimal controls is obtained.     

On the discounted cost of Markovian queuing systems

The paper considers Markovian queues. A rarely discussed characteristic, the discounted cost is analysed. Some continuity properties of the distribution function of he discounted cost are proved. The method used here seems to be applicable to more indicated systems as well.     

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Regular genus: The boundary case

Summary A particular kind of 2-cell imbedding for a class of edge-coloured graphs into surfaces with boundary is introduced and studied. This allows to define, as in [13], where the closed case was traited, a pair of invariants — the regular genus and the hole-number — for every n-manifold with boundary. These invariants are proved to coincide with the classical ones in dimension two, and to be strictly related with a Heegaard-like handlebody decomposition in dimension three. A characterization of 261-1 concludes the work.

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Sunto Nel lavoro si introduce e si studia un nuovo tipo di immersione cellulare in superficie con bordo, per una particolare classe di grafi colorati sugli spigoli.Ciò permette, in analogia con quanto fatto nel caso chiuso in un precedente lavoro [13], di definite per ogni n-varietà con bordo una coppia di invarianti — il genere regolare ed il numero di buchi — che estendono gli omonimi concetti noti per le superficie con bordo. Di tali invarianti si studiano poi relazioni ed interpretazione geometrica, con particolare riguardo alla dimensione tre, dove essi sono collegati ad uno speciale spezzamento tipo-Heegaard.Una caratterizzazione di _n conclude il lavoro.

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Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy). The author was supported by the M.P.I. (Progetto Geometria delle varieta differenziabili).     

On the stable reconstruction of controls in nonlinear distributed systems

This paper deals with the problem of the dynamic approximation of the disturbances acting on a distributed system. A solution algorithm which is stable with respect to informational noise and computing errors is proposed. Examples are given to illustrate the basic constructions.     

On the null-controllability of linear discrete-time systems with restrained controls

This paper is devoted to the null-controllability problem for linear, autonomous, discrete-time systems with restrained controls, i.e., systems described by the equationx _k+1=Ax _k+u _k,u _k , where 0 , but is not necessarily an interior point of . Some criteria for local and global null controllability are given with proofs. These results can be considered as extensions to discrete-time systems of the known corresponding criteria for linear, continuous-time systems.     

On the null-controllability of nonlinear delay systems with restrained controls

Sufficient conditions are developed for the null-controllability of the nonlinear delay process (1) $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{) + B(t) u(t) + f(t, x}{}_{\mathrm{t}}\text{, u(t))}$ when the values of the control functions u lie in an m-dimensional unit cube C^m of E^m. Conditions are placed on f which guarantee that if the uncontrolled system $\text{x}\text{?}\text{(t) = L(t, x}{}_{\mathrm{t}}\text{)}$ is uniformly asymptotically stable and if the linear control system x(t) = L(t, x_t) + B(t) u(t) is proper, then (1) is null-controllable.     

On boundary controllability of one-dimensional vibrating systems

In this paper boundary controllability of one-dimensional vibrating system such as the vibrating string or the vibrating beam is studied. In particular we are concerned with the question whether it is possible to transfer a given initial state of vibration into rest within a given time such that the system stays in rest when the control is turned off. This problem is rephrased as a typical trigonometric moment problem which is solved within the framework of an abstract moment problem in a Hilbert space. The results of null-controllability which are obtained are substantially based on classical results of Ingham and Redheffer concerning trigonometric inequalities and incompleteness of certain sequences of trigonometric functions, respectively. The representation of the general statements follows closely the lines of a paper of Russell. Besides a special case is treated where explicit representations of boundary controls can be given that transfer the system to a permanent rest position. This special case includes amplitude boundary control of the vibrating string and the freely supported beam.     

On the optimal controls of a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions

Summary In this paper we consider the question of existence of optimal controls for a class of systems governed by second order parabolic partial delay-differential equations with first boundary conditions and with controls appearing in the coefficients. In Theorems2.2 and2.3 we present existence and uniqueness of solutions of the first boundary problems. In Theorems3.1 and3.2 we prove that whenever the coefficients of the system converge in the w*-topology (L¹ topology on L^∞) the corresponding solutions converge weakly in an appropriate Sobolev space. Using these basic results we present two theorems (Theorems4.1 and4.2) on the existence of optimal controls. Entrata in Redazione il 21 gennaio 1978.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

The numerical solution of unstable two point boundary value problems

A number of workers have tried to solve, numerically, unstable two point boundary value problems. Multiple Shooting and Continuation Methods have been used very successfully for these problems, but each has weaknesses; for particularly unstable problems their success may be partial. In this paper we develop an algorithm that attempts to solve these problems in a routine manner.The algorithm uses a combination of Multiple Shooting and Range Extension in such a way that the advantages of both are maintained while the effects of their disadvantages are reduced considerably. The success of the algorithm is demonstrated on some particularly unstable problems.     

On the existence of optimal controls for nonlinear infinite dimensional systems

We consider nonlinear systems with a priori feedback. We establish the existence of admissible pairs and then we show that the Lagrange optimal control problem admits an optimal pair. As application we work out in detail two examples of optimal control problems for nonlinear parabolic partial differential equations.     

Minimum principle for systems with boundary conditions and cost having Stieltjes-type integrals

Pontryagin's minimum principle for nonlinear differential systems with Stieltjes boundary constraints and performance index with an integral of the Lebesgue-Stieltjes type is derived. The case of infinite time is also considered.     

Bounds for the additional cost of near-optimal controls

Near-optimal controls are considered for singular problems with a constrained control. These controls result in a higher cost than the optimal cost. Bounds for the additional cost are derived for problems with fixed terminal time or free terminal time and for minimal time problems. An illustrative example is solved of an optimal evasive control of an aircraft against a homing missile.