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ICM 2006

Posters

Abstracts

Section 17Control Theory and Optimization

ICM 2006 – Posters. Abstracts. Section 17

Chain transitive sets for flows on flag bundles

Carlos J. Braga Barros*, Luiz A. B. San Martin

Departamento de Matematica , Universidade Estadual de Maringa, Av. Colombo5790, Maringa - PR, [email protected]

We study the chain transitive sets and Morse decompositions of flows onfiber bundles whose fibers are compact homogeneous spaces of Lie groups.The emphasis is put on generalized flag manifolds of semi-simple (and reduc-tive) Lie groups. In this case an algebraic description of the chain transitivesets is given. Our approach consists in shadowing the flow by semigroups ofhomeomorphisms to take advantage of the good properties of the semigroupactions on flag manifolds. The description of the chain components in theflag bundles generalizes a theorem of Selgrade for projective bundles withan independent proof.

References

[1] Braga Barros, C.J., L.A.B. San Martin, Chain control sets for semigroup ac-tions, Mat. Apl. Comp., 15 (1996), 257–276.

[2] Braga Barros, C.J., L.A.B. San Martin, On the action of semigroups in fiberbundles, Mat. Contemp., 13 (1997), 1–19.

[3] Conley C., Isolated invariant sets and the Morse index, CBMS Regional Conf.Ser. in Math., 38, American Mathematical Society, (1978).

[4] Selgrade, J., Isolate invariant sets for flows on vector bundles, Trans. Amer.Math. Soc. , 203 (1975), 259–390.

[5] San Martin, L.A.B., Maximal semigroups in semi-simple Lie groups, Trans.Amer. Math. Soc., 353 (2001), 5165–5184.

ICM 2006 – Madrid, 22-30 August 2006 1


Frechet differentiability for an optimal control problem oftemperature in thin fabric sheets

Karlis Birgelis

Institute of Mathematics and Computer Science, 29 Raina Blvd., [email protected]

2000 Mathematics Subject Classification. 49K20, 49K22

In paper [1] was considered situation when a moving sheet of glass fabricwas heated up in a furnace in order to burn out oil. The main concern therewas to find the best way of cooling of the fabric sheet after it had beenheated up and oil had been burned out.

We consider a simplified situation from [1], when oil burnout is neglectedand process is steady in time. Our aim is to find the optimal (in some sense)temperature distribution T in fabric sheet Ω = [−l1, l1]×[−l2, l2]×[−δ, δ] byvarying temperature Tht on the heaters Σht of the furnace. As simultaneousconductive-radiative heat transfer occurs in the furnace, then this leads toan optimal control problem for an elliptic boundary value problem with anintegral equation defined on the boundary ([2]):

As thickness 2δ of the sheet is small, temperature T (x1, x2, x3) can beapproximated by a function T (x1, x2) and the original optimal control prob-lem can be replaced by a simpler one:

I(T ) 7→ min,

G1(|T |3T ) + kT = G2(|Tht|3Tht) + kg,

where G1 : L∞(Q) 7→ L∞(Q), G2 : L∞(Σht) 7→ L∞(Q) are linear boundedoperators, g ∈ L∞(Q) and Q = [−l1, l1]× [−l2, l2].

The main result, that we want to present here, is that under certain con-ditions the last equation have one and only one solution Φ(Tht) ∈ L∞(Q)for every fixed Tht ∈ L∞(Σht). Moreover, the mapping Tht 7→ Φ(Tht) fromL∞(Σht) to L∞(Q) is Frechet differentiable and therefore the standard for-mula for increment of the cost functional holds:

I(Φ(T 2ht))− I(Φ(T 1

ht)) = L[T 1ht](T

2ht − T 1

ht) + o(‖T 2ht − T 1

ht‖L∞(Σht)),

where T 1ht ∈ L∞(Σht), T 2

ht ∈ L∞(Σht), L[T 1ht] ∈ L∗∞(Σht).

2 ICM 2006 – Madrid, 22-30 August 2006


References

[1] Buikis, A., Fitt, A. D., A mathematical model for the heat treatment of glassfabric sheets, IMA Journal of Mathematics Applied in Business and Industry10 (1999), 55–86.

[2] Laitinen, M., Tiihonen, T., Conductive-radiative heat transfer in grey materials.Report B6/2000, University of Jyvaskyla, Jyvaskyla, 2000.



A mathematical analysis of an optimal control problem for ageneralized Boussinesq model for viscous incompressible flows

J. L. Boldrini*, E. Fernandez-Cara, M. A. Rojas-Medar

Departamento de Matematica, Universidade Estadual de Campinas,Unicamp-IMECC, CP 6065, 13083-859 Campinas, SP, [email protected]

2000 Mathematics Subject Classification. 49J20, 76D03

We consider an optimal control problem governed by a system of highlynonlinear partial differential equations modeling a viscous incompressibleflow submitted to variations of temperature. The equation for the flow ve-locity was obtained by using a generalized Boussinesq approximation, whichallows a temperature dependent viscosity, [2]; the control variable acts onthe equation for the temperature as an interior source term; the functionalto be minimized is a quadractic functional.

We prove the existence of an optimal solution for this problem; we alsoobtain the associated first order optimality conditions of Pontriagyn typeby using the formalism due to Dubovitskii and Milyutin, [3].

Details of these results and its corresponding proofs the can be foundin [1]

References

[1] J.L. Boldrini, E. Fernandez-Cara, M.A. Rojas-Medar, A Mathematical Analysisof an Optimal Control Problem for a Generalized Boussinesq Model for ViscousIncompressible Flows, Research Report No. 53/05, Unicamp-IMECC, 2005.

[2] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press,Cambridge, 1981.

[3] I.V. Girsanov, Lectures on mathematical theory of extremum problem., Lec-tures notes in Economics and mathematical systems, 67, Springer-Verlag,Berlin, 1972.



Characterizing bilevel problems with extreme point optimalsolutions

Herminia I. Calvete*, Carmen Gale

Dpto. de Metodos Estadısticos, Universidad de Zaragoza, Pedro Cerbuna 12,50009 Zaragoza, [email protected]; [email protected]

2000 Mathematics Subject Classification. 90C30, 90C26

Bilevel programming has been proposed for dealing with decision processesinvolving two decision-makers with a hierarchical structure. They are char-acterized by the existence of two optimization problems in which the con-straint region of the first level problem is implicitly determined by thesecond level optimization problem. In this paper we focus on general bilevelproblems over polyhedra, with first-level constraints involving second-levelvariables. Using the common notation in bilevel programming they can bestated as:

maxx1,x2

f1(x1, x2),

subject to: A1x1 + A2x2 ≤ bx1 ≥ 0

where x2 solves

maxx2

f2(x1, x2)

subject to: B1x1 + B2x2 ≤ dx2 ≥ 0

where x1 ∈ Rn1 and x2 ∈ Rn2 are the variables controlled by the first leveland the second level decision maker, respectively; f1, f2 : Rn −→ R, n =n1 + n2; Ai are m1 × ni-matrices; Bi are m2 × ni-matrices; b ∈ Rm1 andd ∈ Rm2 . We assume that the polyhedra defined by first level and secondlevel constraints are compact. Due to their structure, bilevel programs arenonconvex and quite difficult to deal with. Bard [1] and Dempe [2] are goodgeneral references on this topic.

If f1 and f2 are quasiconcave functions, we prove that the feasible re-gion is comprised of faces of the polyhedron defined by the whole set ofconstraints, S. Hence, the bilevel problem is equivalent to minimizing aquasiconcave function over faces of a polyhedron and so an extreme pointof S exists which solves the problem. Moreover, if the family of functionsf1(x1, x2), f2(x1, x2) verifies that a minimum of the bilevel problem istaken on an extreme point in set S and in all its convex polyhedral subsetsthen f1(x1, x2) and f2(x1, x2) are quasiconcave.



References

[1] Bard, J.F., Practical Bilevel Optimization. Algorithms and applications. KluwerAcademic Publishers, Dordrecht, Boston, London, 1998.

[2] Dempe, S., Foundations of Bilevel Programming. Kluwer Academic Publishers,Dordrecht, Boston, London, 2002.



A local-global classification of Brunovsy linear control systems

M. Carriegos*, A. de Francisco-Iribarren, A. Saez-Schwedt

Departamento de Matematicas, Universidad de Leon, Campus de Vegazana s/n,Leon, [email protected]

2000 Mathematics Subject Classification. 93B25

This work deals with the local study of linear control systems with scalars ina commutative ring, wich is a powerful abstract tool to model delay systems,digital control systems and parameter-depending systems. We recall thatthe problem of classifying linear controllable systems by the feedback equiv-alence was solved by Brunovsky in [1]: a complete set of invariants are theKronecker indices of pairs of matrices, and a canonical form is Brunovsky’scanonical form. With this in mind, we define and study Brunovsky and lo-cally Brunovsky linear systems, continuing the work started in [2]. We givea complete set of invariants for both classes of systems, consisting of lo-cal and global invariants (see [3]). We illustrate our solutions with explicitexamples on the real unit circle and on the real unit sphere. Finally, weoutline a generalization of these results for bcs rings (see [4]). This class ofrings is well known in Control Theory and contains both one dimensionalrings and semi-local rings.

References

[1] Brunovsky, P.A., A classification of linear controllable systems, Kybernetika 3(1970), 173–187.

[2] Carriegos, M., Hermida-Alonso, J.A., Sanchez-Giralda, T., The pointwise feed-back relation for linear dynamical systems, Linear Algebra Appl. 279 (1998),119–134.

[3] Carriegos, M., On the local-global decomposition of linear control systems,Communications in Nonlinear Science and Numerical Simulations 9 (2004),149–156.

[4] Vasconcelos, W.V., Weibel, C.A., Bcs rings, J. Pure Appl. Algebra 52 (1988),173–185.



Structural control of transient waves in strongly nonlinearreaction-diffusion systems

Shmuel Einav1, Lev Eppelbaum* 2, Victor Kardashov1

1Dept. of Biomedical Engineering, Tel Aviv University, Tel-Aviv, Israel; 2Dept. ofGeophysics and Planetary Sciences, Tel Aviv University, Tel Aviv, [email protected]

2000 Mathematics Subject Classification. 74J35

We consider processes which may be described by strongly nonlinear non-stationary equations:

∂u

∂t= Ai(u) (i = 1, 2).

Here Ai(u) is the nonlinear operator of either two types:

A1(u) = 4Φ(u, C, d) + f(u, A, b),

A2(u) =m∑

i=1

∂

∂xiGi

(∂u

∂xi, C, d

)+ h(u, A, b),

where ∆ =∑m

i=1∂2

∂x2i

is the Laplace operator, Φ(s, C, d), Gi(w,C, d), f(s,A, b)and h(s,A, b) are the real-value functions that determine nonlinear diffusionand nonlinear sources intensity; A and C are the real-value parameters de-scribing some inside properties of sources and nonlinear diffusion (structuralparameters); b and d are the external control parameters. The importantproblem is dynamic control of so-called transitional waves (transients) oftwo different structures: shock waves (propagating) and solitons (localized)arising on critical values of the structural parameters. Our main goal is toshow that the transients structure may be controlled by interactive varia-tion of structural and control parameters. On the other hand, it was shownthat by fulfillment of the conditions

Φ(u, C, d) = O(uk), , f(u, a, b) = O(ul) on u → 0 and k > l,

or

Gi(w,C, d) = O(wk), h(u, a, b) = O(ul) on u, w → 0 and k > l

the transients have finite localized or periodic structure in opposite to semi-linear models that admit only transients with asymptotic structure. Thegeneral equation of this types occurs, for instance, in mathematical biologyas models of population dynamics, in chemical physics, etc. Similar models



are used in geophysics for investigation of heat waves propagation in mediawith nonlinear diffusion and nonlinear spatial sources [4]. The role of non-linear diffusion in formation of the finite localized transients was presentedin [2].

Systems with finite localized transients were considered by many au-thors. Promarily is necessary to note [1] where for the first time were con-sidered finite localized solutions of solitons type – compactons. An existenceof the finite localized shock waves and effect of finite localization and peri-odicity of the stationary transitive waves for general evolutional equationsfor the first time were considered in [5]. In present work is suggested aconstructive approach to construction of the finite localized and periodictransients by using of nonlinear diffusion and sources structure interactionwith external control parameters.

We propose to use these equations as models of dynamic processes inbiomedicine (like as self-sustaining pressure pulse in arteries and heartbeatspropagation [3]) and in geophysics for description of the heat localizationin the porous media.

References

[1] Kardashov V.R., Finite control of nonlinear unsteady-state singular processes,Nonlin. Anal.: Theory and Appl. 38 1999, 361–374.

[2] Kardashov, V.R., Eppelbaum, L.V. and Vasilyev O.V., A role of nonlinearterms in Geophysics, Geophys. Res. Lett. 27 No. 14 2000, 2069–2072.

[3] Kardashov, V.R. and Einav, Sh., A structural approach to control of transi-tional waves of nonlinear diffusion-reaction systems: theory and applications tobio-medicine. Intern. J. of Nonl. Discrete Dynamics in Nature and Society 7(1) 2002, 27–40.

[4] Natale, G., and Salusti, Transient solutions for temperature and pressure wavesin fluid-saturated porous rocks, Geophys. J. Int. 124 1996, 649–656.

[5] Rosenau, P. and Hyman, J., Compactons: Solitons with finite wavelength, Phys.Rev. Lett. 70 No.5 1993, 564–567.



A Maple interface for computing variational symmetries inoptimal control

Paulo D. F. Gouveia*, Delfim F. M. Torres

Department of Mathematics, University of Aveiro, 3810-193 Aveiro, [email protected]; [email protected]

2000 Mathematics Subject Classification. 49-04, 49K15, 49S05

The concept of variational symmetry entered into optimal control in theseventies of the twentieth century. Variational symmetries, which keep anoptimal control problem invariant, are very useful in optimal control, butunfortunately their study is not easy, requiring lengthy and cumbersomecalculations. Recently there has been an interest in the application of Com-puter Algebra Systems to the study of control systems, and collectionsof symbolical tools are being developed to help on the analysis and solu-tion of complex problems. The first computer algebra package for comput-ing the variational symmetries in the calculus of variations, and respectiveNoether’s first integrals, was given by the authors in [1]; then extended tothe more general setting of optimal control [2] and, more recently, upgradedin [3] with the introduction of new capacities, by means of several optionalparameters, and improvements of efficiency.

Here we provide a graphical user interface to our computer algebra pack-age [3]. This application is named octool and was created with Maplettechnology, the graphical programming language of the Maple 10 system.With this interface users can, in a point-and-click environment, interactwith all the symbolical tools of the package and deal with concrete prob-lems of optimal control: (i) with the procedure Symmetry, to obtain thevariational symmetries; (ii) with the procedure Noether, to obtain the cor-respondent conservation laws; and (iii) with the PMP (Pontryagin MaximumPrinciple), to try to obtain the Pontryagin extremals or, alternatively, theequations of the Hamiltonian system, stationary condition or the Hamilto-nian. We refer the reader to [2, 3] for a general overview on these Maple pro-cedures. The Maplet octool allow us to quickly investigate the problems,without learning all the optional parameters of the Maple procedures [1]–[3]. Moreover, it permits additional algebraic manipulations. The completeMaple package, with the new procedure octool, can be freely obtained fromhttp://www.mat.ua.pt/delfim/maple.htm together with many practicalexamples and an online help database for the Maple system.



References

[1] Gouveia, P.D.F., Torres, D.F.M., Computacao Algebrica no Calculo dasVariacoes: Determinacao de Simetrias e Leis de Conservacao (in Portuguese),TEMA Tend. Mat. Apl. Comput. 6 (2005), No. 1, 81–90.

[2] Gouveia, P.D.F., Torres, D.F.M., Automatic Computation of ConservationLaws in the Calculus of Variations and Optimal Control, Comput. MethodsAppl. Math. 5 (2005), No. 4, 387–409. [MR2194205] [Zbl 1079.49019]

[3] Gouveia, P.D.F., Torres, D.F.M., Rocha, E.A.M., Symbolic Computa-tion of Variational Symmetries in Optimal Control, Control & Cy-bernetics (Accepted. To appear, presumably, on No. 3/2006). E-Print:http://pam.pisharp.org/handle/2052/113.



Evaluating a multiple quality characteristic process using dataenvelopment analysis

Ester Gutierrez*, Sebastian Lozano

Department of Industrial Managment, Escuela Superior de Ingenieros, Caminode los Descubrimientos s/n 41092, Sevilla, [email protected]; [email protected]

2000 Mathematics Subject Classification. 90c90, 90c05, 82C32

Taguchi methods provide a comprehensive methodology for quality im-provement. The objective is based on a robust design and involves con-ducting experiments using orthogonal arrays and the estimation factor lev-els combination that optimizes a given performance measure, typically aSignal-to-Noise ratio. The problem is more complex in the case of multipleresponses since the combinations of factor levels that optimise the differentresponses usually differ.

In this paper, a three-step approach to find the optimal parameter com-bination in robust design is presented. Starting the results of the exper-iments, their Mean Square Deviations (MSD) are used to train a NeuralNetwork which afterwards can be used to estimate the MSD of those factorcombinations for which no experimental data exist. Once MSD of all qualitycharacteristics for all factor combinations are available, a units-invariant,non-radial, pure-input, Variable Returns to Scale -Data Envelopment Anal-ysis (DEA) model is proposed to assess their relative efficiency. Finally, asecond DEA model is used to choose among the efficient factor combina-tions the one that allows a higher overall penalization of the Quality Lossassociated to the different characteristics. The proposed approach is appliedto a number of case studies taken from the literature and compared withexisting approaches.

References

[1] Caporaletti,L.E., Dula, J.H. and Womer, N.K. Performance evaluation basedon multiple attributes with nonparametric frontiers, Omega. 27 (1999), 637–645.

[1] Liao, H-C.,Using N-D method to solve multi-response problem inTaguchi”,Journal of Intelligent Manufacturing. 16 (2005),331–347.

[2] Liao, H-C., A data envelopment analysis method for optimizing multi-responseproblem with censored data in the Taguchi method, Computers and IndustrialEngineering. 46 (2004), 817–835.



[3] Liao, H-C.,Chen, Y-K. Optimizing multi-response problem in the Taguchimethod by DEA based ranking method,International Journal of Quality andReliability Management. 7 (2002), 825–837.



Behavioral approach for model reduction and approximation

Ha Binh Minh

Department of Mathematics, University of Groningen, The [email protected]

2000 Mathematics Subject Classification. 93A30, 93B11, 93B15, 93C05,93C15

The research described in this poster deals with reduction and approxi-mation of mathematical models for linear dynamical systems. Given themodel with a certain complexity, the problem will be investigated of ap-proximating this model by a lower order, less complex one in such a waythat the lower model model retains or closely approximates the behavior ofthe original model. The distinctive feature of this research is that it intendsto study the model reduction problem from a behavioral point of view. Thepurpose is to establish a representaion-free approach to model reductionand approximation, one which considers the system itself as the startingpoint, instead of one of ots particular representation. We will also avoidthe use of input/output partitions of the system variables. The advantagesof this approach are wider variety of model classes that can be considered,and the flexibility with which the results can be adapted to the particularsystem representation.

References

[1] Willems, J. C.; Trentelman, H. L. On quadratic differential forms. SIAM J.Control Optim. 36 (1998), no. 5, 1703–1749

[2] Antoulas, A. C. A new result on passivity preserving model reduction. SystemsControl Lett. 54 (2005), no. 4, 361–374.

[3] Sasane, Amol J. Distance between behaviours. Internat. J. Control 76 (2003),no. 12, 1214–1223.



Optimal control problem of some differential parabolic inclusion– convergence of Galerkin approximation

Andrzej Just

Center of Mathematics and Physics, Technical University of Lodz, 90-924 Lodz,Al. Politechniki 11, [email protected]

2000 Mathematics Subject Classification. 49J24, 49J40, 49M15

The paper is concerned with optimization problem and its approximationfor the cost functional.

J(u) = λ1 ‖ y − yd ‖2L2(0,T ;L2(Ω)) +λ2 ‖ u− u0 ‖2

L2(0,T ;L2(Ω))

λ1λ2 > 0, λ1 + λ2 > 0

where y = y(u) is a solution of parabolic inclusion

∂y(t, x)∂t

−n∑

i,j=1

∂

∂xi

(aij(x)

∂y(t, x)∂xj

)+a0(x)y(t, x)+∂χ(y(t, x)) u(t, x) a.e. Q

y(0, x) = y0 a.e. Q

Ω ⊂ Rn is a set of C0class with boundary Γa0, aij ∈ C∞(Ω) for i, j = 1, 2, · · · , n and

∑aij(x)ξiξj > α

n∑i=1

ξi2 ∀ξi, ξj ∈ R

a0(x) > α for certain α > 0, [aij ]16i,j6n is symetric matrix

χ(y(t)) =

0 for y(t) ∈ C+∞ for y(t) ∈ H1

0 (Ω)\C

The set C is any convex closed subset of H10 (Ω), int C 6= 0 and y0 ∈ int C.

We derive some results on the existence of optimal solutions. We describethe Galerkin approximation and we demonstrate of the weak condensationpoints of a set of solutions of the approximate optimization problems. Eachof these points is a solutions of the initial optimization.



References

[1] A. Just. (2003) Existence Theorems and Galekin Approximation for Non-LinearEvolution Control Problems, Optimization, Vol.52, No.3, 287-300.

[2] A. De binska-Nagorska, A. Just and Z. Stempien. (1998) Analysis and Semidis-crete Galerkin Approximation of a Class of Nonlinear Parabolic Optimal Con-trol Problems, Computers Math. Applic. Vol.35(6), 95-103.

[3] F.Troltzsch. (1994) Semidiscrete Ritz-Galerkin Approximation of Non-linearParabolic Boundary Control Problems-Strong Convergence of Optimal Con-trols, Appl. Math. Optim. Vol.29(3), 309-329.



Static versus dynamic feedback equivalence

M. M. Lopez-Cabeceira*, M. T. Trobajo

Departamento de Matematicas, Universidad de Leon, 24071-Leon, [email protected]

2000 Mathematics Subject Classification. 93B25; 93B52; 13F99

An m-input n-dimensional linear dynamical system Σ = (A,B) over a com-mutative ring R is a pair of matrices with entries in R, where A is an n×nmatrix and B is an n×m matrix. Σ is called (statically) feedback equivalentto Σ′ = (A′, B′) if there exist invertible matrices P and Q, and a feedbackmatrix F such that B′ = PBQ and A′ = P (A + BF )P−1 (or equivalentlyPA−A′P = B′K for some matrix K). The feedback classification problem(i.e. to obtain a complete set of invariants that characterizes the feedbackequivalence class of Σ) is wild. In [1] and [5] can be found basic terminologyan properties on linear sistems and the feedback action.

One of the main difficulties of the feedback equivalence problem is tofind that invertible matrices P and Q. Nevertheless, this difficulty can beeluded, in some cases, by means of an alternative action, which enlargesthe field of invertible matrices: the dynamic feedback action. It consists ofapplying the static one to extended systems Σ(r) of the following form

Σ(r) =

((0

A

),

(Idr

B

)).

Static feedback implies dynamic feedback, but the converse is not truein general. The open question is: When are dynamic and static feedbackequivalent?

In this poster we characterize the class of commutative rings in whichthe static and dynamic feedback actions are equivalents: stable rings (see [3]and [4]). The characterization is based on feedback invertibility problems(see [2]).

References

[1] Brewer, J. W., Bunce, F. S., Van Vleck F. S. Linear Systems over CommutativeRings. Marcel Dekker, New York (1986).

[2] Carriegos, M., Garcıa Planas, I., On matrix inverses modulo a subspace. LinearAlgebra Appl. 379 (2004) 229-237.

[3] Estes, D., Ohm, J. Stable range in commutative rings. J. Algebra 7 (3) (1967)343-362.



[4] Hermida-Alonso, J. A., M.M. Lopez-Cabeceira , M.T. Trobajo When are dy-namic and static feedback equivalent?. Linear Algebra Appl. 405 (2005) 74-82.

[5] Hermida-Alonso, J. A. On linear algebra over commutative rings, In Handbookof Algebra, vol. 3, Elsevier Science (2003) 3-61.



Optimal design of the support of the control for the 2-D waveequation

Arnaud Munch

Laboratoire de Mathematiques, UMR CNRS 6623, Universite de Franche-Comte,16 route de Gray 25030, Besancon, [email protected]

2000 Mathematics Subject Classification. 35L05, 49J20, 65K10, 65M60,93B05

Let us consider a rectangular bounded domain Ω ∈ R2 and a subset ωof positive Lebesgue measure |ω|. For any time T > T ?(Ω\ω) and anyinitial data (y0, y1) ∈ H1

0 (Ω) × L2(Ω), there exists a distributed controlvω ∈ L2(ω × (0, T )) such that the unique solution y ∈ C([0, T ];H1

0 (Ω)) ∩C1([0, T ];L2(Ω)) of

ytt −∆y = vωXω in Ω× (0, T ),

y = 0 on ∂Ω× (0, T ), (y(·, 0), yt(·, 0)) = (y0, y1) in Ω,(1)

satisfies y(., T ) = yt(., T ) = 0 in Ω [3]. Xω denotes the characteristic functionof ω. Following [1] in a similar context, we address the numerical resolutionof the problem (Pω) : infω⊂ΩL

J(Xω) = 1/2||vω||2L2(ω×(0,T )) where ΩL =ω ∈ Ω; |ω| ≤ L|Ω| which consists in finding the optimal shape of ω ∈ ΩL

in order to minimize the L2-norm of the corresponding control vω. In orderto apply a gradient descent algorithm and assuming ω ∈ C1(Ω), we firstdetermine the so-called shape derivative of J independently of any adjointsolution. Expressed as a curvilinear integral on ∂ω × (0, T ), the derivativethen permits to define a descent direction and build a decreasing sequenceof domains for J . Numerical experiments in the framework of the levelset method [2] highlight the influence of the data, particularly the time Ton the optimal domain. Very interestingly, the method permits to obtainthe control of minimal L2-norm supported on a domain ω of arbitrarilysmall Lebesgue measure and which drives the system (1) to rest after anarbitrarily small time T . We also investigate the well-posedness characterof the problem (Pω) by considering its convexification (CPω) where theinfimum over the set L∞(Ω, 0, 1) of characteristics functions is replaced bythe infimum over the convex envelop set L∞(Ω, [0, 1]) of densities functions[5]. Several experiments indicate that problems (Pω) and (CPω) coincideand permit to conjecture that the original problem is always well-posed(the infimum is reached in the set of characteristics functions). We refer to[4] for detailed statements and numerical simulations.



References

[1] Asch M., Lebeau G., Geometrical aspects of exact controllability for the waveequation - A numerical study, Esaim : Cocv, 3 (1998), 163–212.

[2] Burger M., Osher S.J., A survey on level set methods for inverse problemsand optimal design, European Journal of Applied Mathematics, 16(2) (2005),263–301.

[3] Haraux A., A generalized internal control for the wave equation in a rectangle.,J. Math. Anal. Appl. , 153 (1990), 190–216.

[4] Munch A., http://www-math.univ-fcomte.fr/amunch/

[5] Pedregal P., Vector variational problems and applications to optimal design,Esaim : Cocv, 11 (2005), 357–381.



Control results for morphological equations

Jose Alberto Murillo Hernandez

Departamento de Matematica Aplicada y Estadıstica, Universidad Politecnica deCartagena, Paseo de Alfonso XIII, s/n 30203-Cartagena, [email protected]

2000 Mathematics Subject Classification.

Families of time-evolving sets (usually called tubes) arise in many frame-works: dynamical economic models, image processing, shape optimization,visual control or propagation of fronts, among others. Morphological equa-tions, introduced by J.-P. Aubin in the nineties (see [1] and referencestherein) as a particular case of mutational equations, provide a tool fordescribing the dynamics of tubes by means of a suitable notion of veloc-ity (properly, a set of velocities) inspired by shape derivative (see [4]). Inthis communication we discuss different questions about controllability andstabilization of morphological equations. We also consider the case of jointevolutionary-morphological systems (see [1], [3], [5]), composed of a dif-ferential inclusion (governing the evolution of trajectories) coupled witha morphological equation (ruling the evolution of the sets of constraints).Examples and applications of theoretical results will be provided.

References

[1] Aubin, J.-P., Mutational and Morphological Analysis. Tools for Shape Evolutionand Morphogenesis. Birkauser, Boston, 1999.

[2] Aubin, J.-P., Frankowska, H., Set-Valued Analysis. Birkauser, Boston, 1990.

[3] Aubin, J.-P., Murillo, J.A., Morphological Equations and Sweeping Processes.In Nonsmooth Mechanics and Analysis. Theoretical and Numerical Advances(ed. by P. Alart, O. Maisonneuve and R.T. Rockafellar). Advances in Mechanicsand Mathematics 12, Springer, New York 2006, 249–259.

[4] Delfour, M.C., Zolesio, J.-P., Shapes and Geometries. Analysis, DifferentialCalculus and Optimization. SIAM, 2001.

[5] Murillo, J.A., Tangential Regularity in the Space of Directional-MorphologicalTransitions, J. Convex Anal. 13 (2006), to appear.



Using maximal cover inequalities as cutting planes in 0-1problem solving

S. Munoz

Departamento de Estadıstica e Investigacion Operativa I, UniversidadComplutense de Madrid, Facultad de Ciencias Matematicas, CiudadUniversitaria, 28040 Madrid, [email protected]

2000 Mathematics Subject Classification. 90C10, 90C05

We present an efficient procedure for identifying all maximal covers fromthe set of covers implied by a 0-1 knapsack constraint, see [5] for furtherexplanation.

In easy terms, a cover can be considered as a subset of indices of 0-1 variables where at most k of such variables can take the value 1. Inparticular, we are interested in the so-called maximal covers from the set ofcovers implied by a 0-1 knapsack constraint, i.e., covers derived from a 0-1knapsack constraint such that the inequalities induced by any other coversthat can be derived from the knapsack constraint are not tighter than theirinduced inequalities; see [3] for more details.

It is well known that inequalities induced by covers have numerous appli-cations in 0-1 model tightening, see [1], [2] and [4] among others. Therefore,our procedure can be very useful for this purpose.

Some computational experiments are reported for single source capac-itated plant location problem instances drawn from the literature. Theyshow the efficiency of treating the inequalities induced by our maximal cov-ers as cutting planes in a branch-and-cut framework.

References

[1] Crowder, H., Johnson, E. L., Padberg, M., Solving Large-Scale Zero-One LinearProgramming Problems, Oper. Res. 31 (1983), 803–834.

[2] Escudero, L. F., Munoz, S., On Characterizing Tighter Formulations for 0-1Programs, European J. Oper. Res. 106 (1998), 172–176.

[3] Escudero, L. F., Munoz, S., On Characterizing Maximal Covers, InvestigacionOper. 23 (2002), 136–149.

[4] Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., Progress in Lin-ear Programming-Based Algorithms for Integer Programming: An Exposition,INFORMS J. Comput. 12 (2000), 2–23.

[5] Munoz, S., On Identifying Maximal Covers, SIAM J. Discrete Math. 18 (2005),749–768.



Behavioral approach for control systems of PDE

Diego Napp

Department of Mathematics, University of Groningen, The [email protected]

I will present a theory of control for distributed systems (i.e those definedby systems of constant coefficient partial differential operators) via the be-havioral approach of Willems. In this approach a system is defined as atriple Σ = (A, q,B) where A is the signal set, q is an integer, and B ⊂ Athe set of admissible signal vectors. I will show the relation between be-haviors of distributed systems and submodules of free modules over thepolynomial ring in several indeterminates. And finally important conceptssuch as controllability and observability will be defined in such a context.

References

[1] Willems, Jan C. Paradigms and puzzles in the theory of dynamical systems,IEEE Trans. Automat. Control, 36, 1991

[2] Pillai, Harish K. and Shankar, Shiva, A behavioral approach to control of dis-tributed systems, SIAM J. Control Optim., SIAM Journal on Control andOptimization, 37, 1999,

[3] Oberst, Ulrich, Multidimensional constant linear systems, Acta Appl. Math.,Acta Applicandae Mathematicae. An International Survey Journal on ApplyingMathematics and Mathematical Applications, 20, 1990,



How do the algorithms of maximum likelihood parameters’estimation affect on Stochastic Volatility Models?

Pilar Munyoz, Rouhia Noomene*

Department of Statistics and Operations Research Technical University ofCatalonia, [email protected]

The objective of this paper is to study how algorithms of optimizationaffect the parameters’estimation of Stochastic Volatility models (SV). In ourresearch we have represented the SV models in linear state space form andapplied the Kalman Filters to estimate the different unknown parameters’of the model.Many methods have been proposed by researchers for the estimation ofthe parameters in the case of the linear state space models. In our workwe have emphasized on the estimation through the Maximum Likelihood(ML). Statisticians have used many algorithms to optimise the likelihoodfunction and they have proposed many filters; publishing their results inmany papers. In spite of the fact that this field is so extended, we haveemphasized our study in the financial field. Two quasi-Newton algorithms:Berndt, Hall, Hall, and Hausman (BHHH) and Broyden-Fletcher-Goldfarb-Shanno (BFGS), and the Expectation-Maximization (EM) algorithm havebeen chosen for this study. A practical study of these algorithms applied tothe maximization of likelihood by means of the Kalman Filter have beendone. The results are focused on efficiency in time of computation andprecision of the unknown parameters’ estimation.A simulation study has been carried out, using as true values the parametersof this model published in the literature[1], in order to test the efficiencyand precision of our implemented algorithms. Finally we have applied themto the real series IBEX 35 stock returns.Key words: State space model, Kalman filer, maximum likelihood, BHHH,BFGS and EM.

References

[1] Shumway, R. H., Stoffer, D. S. , Time Series Analysis and Its Applications(ed.by Springer Texts in Statistics)Hardcover.



Optimal design of the damping set for the stabilization of thewave equation

Arnaud Munch, Pablo Pedregal, Francisco Periago*

Departamento de Matematica Aplicada y Estadıstica, Universidad Politecnica deCartagena, Paseo Alfonso XIII, 30203 Cartagena, [email protected]

2000 Mathematics Subject Classification. 35L05, 49J20, 49J45, 65K10

We consider the problem of optimizing the shape and position of the viscousdamping set for the internal stabilization of the linear wave equation indimensions 1 and 2. Several works [1, 2] have showed that this kind ofproblems may be ill-posed.

This paper aims at giving a well-posed relaxed formulation for this non-linear optimal design problem. Precisely, in a first theoretical part, follow-ing [5], we reformulate the problem into an equivalent non-convex vectorvariational one by using a characterization of divergence-free vector fields.Then, by means of gradient Young measures [4], we obtain a relaxed for-mulation in which the original cost density is replaced by its constrainedquasi-convexification. This implies that the relaxed problem is well-posedin the sense that there exists a minimizer and, in addition, the infimum ofthe original problem coincides with the minimum of the relaxed one. In asecond numerical part, we address the resolution of the relaxed problem byusing a first order gradient descent method. We present several numericalexperiments which show that for small and constant values of the dampingpotential the original problem is well-posed. However, when the potentialexceeds a critical value the original problem is no more well-posed whichjustifies the relaxation’s procedure. We then propose a penalization tech-nique to recover the minimizing sequences of the original problem from theminimizers of the relaxed one. The numerical experiments show the excel-lent performance of this penalization technique.

Detailed statements and complete proofs of the results mentioned inthis abstract may be found in [3].

References

[1] Cox, S.J., Designing for optimal energy absorption II, The damped wave equa-tion, International series of numerical mathematics, 126 (1998), 103–109.

[2] Hebrard, P. and Henrot, A., Optimal shape and position of the actuators forthe stabilization of a string, Systems and control letters 48 (2003), 199–209.



[3] Munch, A., Pedregal, P. and Periago, F., Optimal design of the damping setfor the stabilization of the wave equation, submitted for publication. Preprintavailable at http://matematicas.uclm.es/omeva/ preprints06.php

[4] Pedregal, P., Parametrized Measures and Variational Principles, Birkhauser,1997.

[5] Pedregal, P., Vector variational problems and applications to optimal design,ESAIM: COCV 11 (2005) 357–381.



Hybrid output feedback controls and stabilization of dynamicalsystems

Elena Litsyn, Yurii Nepomnyashchikh, Arcady Ponosov*

Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel; CIGENE- Centre for Integrative Genetics & Department of Mathematics and Informatics,Eduardo Mondlane University, Maputo, Mozambique; CIGENE - Centre forIntegrative Genetics & Department of Mathematical Sciences and Technology,Norwegian University of Life Sciences, P. O. Box 5003, NO-1432 As, Norway

2000 Mathematics Subject Classification. 93D15

Hybrid dynamical systems [1], where discrete and continuous dynamics arecoupled together, arise when there is a necessity of combining logical deci-sion with continuous control laws. Many examples of hybrid systems can befound in manufacturing systems, intelligent vehicle highway systems, chem-ical plants. Other examples come from biology and, in particular, fromgenetics.

The problem of how a hybrid output feedback control (HFC) can bedesigned to stabilize a continuous control plant, provides significant math-ematical challenges, especially if the output feedback within the plant failsto stabilize the system. For instance, this may be the case if no completeinformation on the plant’s dynamics is available.

Any control system of differential equations below is assumed to satisfythe conditions of controllability and observability. In (i)-(iii) the controlsystems are linear.

The following results are discussed:(i) An n-dimensional system can be stabilized by an affine HFC with

infinitely many states.(ii) There exist 2-dimensional systems that cannot be stabilized by a

linear HFC with finitely or infinitely many states.(iii) Sufficient conditions on n-dimensional systems admitting a linear

HFC with finitely many states (unpublished). The conditions become nec-essary and sufficient in the 2-dimensional case.

(iv) Sufficient conditions for hybrid output feedback stabilization ofquasilinear control systems.

(v) Stabilization algorithms.

References

[1] Liberzon, D., Switching in Systems and Control. Birkhauser, Boston-Basel-Berlin, 2003.



Computer methods in control synthesis of uncertain systems: aconvex optimization approach

Francesc Pozo

Departament de Matematica Aplicada, Universitat Politecnica de Catalunya,Comte d’Urgell, 187, 08036 Barcelona, [email protected]

2000 Mathematics Subject Classification. 93C10

The last few years witnessed an increasing interest in the problem of controlsynthesis of nonlinear systems. A recently derived stability criterion fornonlinear systems [3] –which has a remarkable convexity property– and thedevelopment of numerical methods for verification of positivity [1] allowsthe computation –via semidefinite programming– of stabilizing controllersfor the case of systems with polynomial or rational vector fields. Using thetheory of semialgebraic sets [2] these computational tools are extended forthe case of polynomial or rational systems with uncertain parameters.

References

[1] Prajna, S., Parrilo, P.A., Rantzer, A., Nonlinear control synthesis by convexoptimization, IEEE Transactions on Automatic Control 49 (2004), 310–314.

[2] Putinar, M., Positive polynomials on compact semi-algebraic sets, Indiana Uni-versity Mathematical Journal 42 (1993), 969–984.

[3] Rantzer, A., A dual to Lyapunov’s stability theorem, Systems and ControlLetters 42 (2001), 161–168.



Design of H∞ control for uncertain systems: an LMI approach

Gisela Pujol

Department of Applied Mathematics III, Escola Universitaria d’EnginyeriaTecnica Industrial de Barcelona, Universitat Politecnica de Catalunya, CompteUrgell 187, 08036 Barcelona, [email protected]

The aim of this work is to study the reliable control design problem foruncertain interconnected systems. The continuous H∞ control problem issolved via elementary manipulations on linear matrix inequalities (LMI). Amore practical model of actuators or control channels failures than outageis adopted. Based on LMI design approach, a class of reliable decentralizedlocal state feedback controllers is presented. Moreover, the approach offersnew potentials for problems that cannot be handled using earlier techniques.We obtain linear parameter LMI constraints, allowing parametric Lyapunovfunctions. The resulting control systems are robustly stable against plantuncertainty and failures.

References

[1] Apkarian, P., Tuan, H.D., Bernussou, J. Continuous-time analysis, eigenstruc-ture assignment and H2 synthesis with enhanced LMI characterizations, Proc.of 39th Conference on Decision and Control, (2000), 1489–1494.

[2] Khargonekar, P., Petersen, I., Zhou, K. Robust stabilization of uncertain linearsystems: quadratic stabilization and H∞ control theory, IEEE Trans. Auto-matic Control, 35(3) (1990), 356–361.

[3] Liao, F., Wang, J.L., Yang, G-H., Reliable Robust Flight Tracking Control: AnLMI Approach. In IEEE Trans. Control Systems Tech., 10(1) (2002), 76–89.

[4] Yang, G.-H., Wang, J.L., Soh, Y.C. Reliable Guaranteed Cost Control for Un-certain Nonlinear Systems, IEEE Trans. Automatic Control, 45(11) (2000),2188–2192.



An algorithm of adaptive ε-minimax control for pursuit-evasionin discrete convex dynamical systems with several pursuers

Andrei F. Shorikov

Department of Information Systems in Economics, Urals State University ofEconomics, [email protected]

2000 Mathematics Subject Classification. 91 A50, 91 A06

In this report we consider the problem of adaptive ε-minimax control for thepursuit-evasion process with incomplete information [1]–[4] in the class ofconvex discrete dynamical systems consisting of several controlled objects.Each object Ij , j ∈ 1, 2, ·, n = 1, n controlled by n pursuers Pj and theobject II controlled by the evader E have a dynamics described by con-vex discrete recurrent vector equations. It is assumed that each pursuer Pj

knows the values of past realizations of his control impact on the object Ij .He also knows past realizations of the available information signal aboutthe object II, which is measured with an error generated by a convex dis-crete vector equation. Moreover, the pursuer Pj is informed about sets thatrestrict the changes of all a priori indeterminate parameters that describethe dynamics of the objects Ij and II, and the input of the correspondinginformation signal. Each of these sets is assumed to be a convex, closed, andbounded polyhedron (with a finite number of vertices) in the correspond-ing Euclidean space. We also assume that there exists a participant P , ageneral coordinator of pursuit, who knows all the information known to thepursuers Pj , j ∈ 1, n, and that, at any moment of time, he can tell to anyof them the values of state vectors of object II. To realize the ε-minimaxpursuit control for any fixed number ε > 0 in a chosen family of admissiblestrategies of adaptive controls, we propose a finite recurrent algorithm, eachstep of which is based on the realization of a process of posterior ε-minimaxnonlinear filtration [3] an on solving some problems of linear and convexprogramming. The results obtained in this report are based on the works[1]–[4] and can be used in the computer modeling of real dynamical processand in the optimal design of navigation and control devices for differenttransportation systems.

This work was supported by the Russian Foundation for Basic research,project no. 04-01-00059.



References

[1] Krasovskii, N. N., Subbotin, A. I. Game-Theoretical Control Problems.Springer, New York, Berlin, 1988.

[2] Kurzhanskii, A. B. Control and Observation under Uncertainty Conditions.Nauka, Moscow, 1977 (in Russian).

[3] Shorikov, A. F. Minimax Estimation and Control in Discrete-Time DynamicalSystems. Ural State University Publisher, Ekaterinburg, 1997 (in Russian).

[4] Ho, Y. C., Bryson, A. E., Baron, S. Differential Games and Optimal Pursuit–Evasion Strategies. IEEE Trans. Autom. Control. 10(4) (1965), 385–389.



Decision method of optimal fuzzy graph applying fuzzy decision

Shuya Kanagawa, Kimiaki Shinkai, Ei Tsuda, Hiroaki Uesu*, Hajime Yamashita,Michiko Yanai

Waseda University, 1-3-1-612 Kamiya Kita-ku Tokyo, [email protected]

2000 Mathematics Subject Classification. 26E50

We could analyze the inexact information efficiently and investigate thefuzzy relation.

By applying the fuzzy graph theory, we propose a fuzzy node fuzzygraph, and we transform it to a crisp node fuzzy graph by using T-normfamily.

In this paper, we present an analysis method of the fuzzy node fuzzygraph. The fuzzy node fuzzy graph is usually too complicated to investigateits structural feature. Therefore, we would propose a transformation methodfrom the fuzzy node fuzzy graph to the crisp node fuzzy graph (the generalfuzzy graph) by applying T-norm family ”quasi-logical product”, and wecould obtain the crisp node fuzzy graph sequence. Next, we defined twofunctions (fuzzy distance function fd(λ) and fuzzy connectivity functionfe(λ) to decide the optimal crisp node fuzzy graph in the crisp node fuzzygraph sequence by applying fuzzy decision.

Moreover, we shall illustrate its practical effectiveness with the casestudy concerning sociometry analysis.

References

[1] H.Uesu, H.Yamashita, Instruction and Cognition Analysis Applying FuzzyNode Fuzzy Graph, 17th Fuzzy System Symposium 5E2-3, Japan Society forFuzzy Theory and Systems, 2001, 221–222 (in Japanese).

[2] H.Uesu, Sociometry Analysis Applying Fuzzy Node Fuzzy Graph, Journal ofJapan Society for Fuzzy Theory and Systems Vol.14, No.3, 2002 (in Japanese).

[3] H.Uesu, H.Yamashita, Connectivity Properties of T-Norm Families and its Ap-plication, Int’l Conference on Computer, Communication and Control Tech-nologies, 2003.

[4] H.Uesu, E.Tsuda, Clustering Level Analysis Applying Fuzzy Theory and itsApplication, Conference of Japan Society for Educational Technology, 2003,695–696 (in Japanese).

[5] H.Uesu, Fuzzy Node Fuzzy Graph, T-Norm and its Application, Journal ofJapan Society for Fuzzy Theory and Intelligent Informatics Vol.16 No.1, 2004,88–95 (in Japanese).



Cost of tracking for differential stochastic equations in Hilbertspaces

Viorica Mariela Ungureanu

Department of Mathematics, “Constantin Brancusi” University , B-dulRepublicii, nr.1, Targu-Jiu, jud. Gorj, 210152, Romania

2000 Mathematics Subject Classification. 93E20, 93C55

We consider the following tracking problem:given a signal r(t) we want tominimize the cost

J(s, u) = limt→∞1

t− sE

t∫s

‖C(σ) (x(σ)− r(σ))‖2 + 〈K(σ)u(σ), u(σ)〉 dσ

in a suitable class of control u subject to the following equation

dx(t) = A(t)x(t)dt+B(t)u(t)dt+m∑

i=1

(Gi(t)x(t) + Hi(t)u(t)) dwi(t)x(s) = x,

where the family A(t), t ∈ [0,∞), generates an evolution operator. The Ric-cati equation associated with this problem (see also [1] for the finite dimen-sional case) is in general different from the conventional Riccati equation(see [3], [4]). Under stabilizability and uniform observability conditions andassuming that the operator K(t), so-called control weight, is uniformly posi-tive definite, we establish that this equation has a unique, positive, boundedon R+ and stabilizing solution. Using this result we find the optimal con-trol and the optimal cost for the tracking problem. Our results are differentto the ones obtained under detectability conditions (see [4]) because weproved in [3] that uniform observability does not imply detectability. In [1]it is considered only the solvability of the backward Riccati equation (BRD)associated to the corresponding LQR problem, unlike our case where it isdiscussed the existence of a global solution of the Riccati equation. We needa global solution because in our situation the control problem is consideredon infinite interval. To obtain the global solution of the Riccati equation,which is a strong limit of solutions of the BRD, we used the algorithmproposed in [1] to prove the existence of these solutions, but we also mustsolve many problems involved by the unboundedness of the coefficients ofthe equations.



References

[1] Chen S.,Zhou,X.Y., Stochastic Linear Quadratic Regulators with IndefiniteControl Weight Cost.II, SIAM J. Control Optim. 39 (2000), 1065–1081.

[2] Pritchard,A. J. ,Zabczyc, J.,Stability and Stabilizability of Infinite DimensionalSystems, SIAM Review 23 (1981), 25–52.

[3] Ungureanu,V. M., Riccati Equation of Stochastic Control and Stochastic Uni-form Observability in Infinite Dimensions,in: Proceedings of the conference“Analysis and Optimization of Differential Systems”, Kluwer Academic Pub-lishers, 2003, 421–423.

[4] Da Prato,G., Ichikawa, A., Quadratic Control for Linear Time-Varying Sys-tems, SIAM. J. Control Optim. 28(1990), 359–381.



Adaptive robust state pbservers for a class of uncertainnonlinear dynamical systems

Hansheng Wu

Department of Information Science, Prefectural University of Hiroshima,Shobara–shi, Hiroshima 727–0023, [email protected]

2000 Mathematics Subject Classification. 93B07, 93C40, 93D21, 93C10

In the robust state observer design problem of dynamical systems with sig-nificant uncertainties, the upper bounds of the vector norms on the uncer-tainties are generally supposed to be known, and such bounds are employedto construct some types of robust state observers [1]. However, in a numberof practical control problems, such bounds may be unknown. Therefore,for such a class of uncertain dynamical systems whose uncertainty boundsare unknown, an adaptive scheme should be introduced to update theseunknown bounds to construct some types of robust state observers. In gen-eral, such an observer is called adaptive robust state observer. In the pastdecades, few efforts are made to consider the problem of adaptive robuststate observers for uncertain dynamical systems with the unknown boundsof uncertainties or perturbations [2].

In this paper, we consider the problem of adaptive robust state observerdesign for a class of nonlinear dynamical systems with significant uncer-tainties. Here, we suppose that the upper bound of the nonlinearity oruncertainty function is a linear function of some parameters which are stillassumed to be unknown. We want to develop a class of continuous adap-tive robust state observers which can guarantee the asymptotic convergenceof the observation error between the observer state estimate and the truestate. For this purpose, we employ an improved adaptation law with σ–modification [3] to estimate the unknown parameters. Then, by making useof the updated values of these unknown parameters we construct a classof adaptive robust state observers. We also show that by employing ouradaptive robust state observer, the observation error can converge asymp-totically to zero in the presence of significant uncertainties. In particular,the adaptive robust state observer with an updating law, proposed in thepaper, is continuous, and can be easily implemented in practical controlproblems. Finally, an illustrative example is given to demonstrate the va-lidity of the results obtained in the paper.



References

[1] Walcott, B.L., Zak, S.H., State Observation of Nonlinear Uncertain DynamicalSystems, IEEE Trans. Automat. Contr., AC–32 (1987), 166–170.

[2] Chen, Y.H., Adaptive Robust Observers for Non–linear Uncertain systems, Int.J. Syst. Sci., 21 (1990), 803–814.

[3] Wu, H., Adaptive Robust Tracking and Model Following of Uncertain Dynami-cal Systems with Multiple Time Delays, IEEE Trans. Automat. Contr., AC–49(2004), 611–616.


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