Zentralblatt MATH Database 1931 – 2010c© 2010 European Mathematical Society, FIZ Karlsruhe & Springer-Verlag

Zbl 1175.34001

Grune, Lars; Junge, OliverOrdinary differential equations. An introduction from the dynamical sys-tems perspective. (Gewohnliche Differentialgleichungen. Eine Einfuhrungaus der Perspektive der dynamischen Systeme.) (German)Vieweg Studium: Bachelorkurs Mathematik. Wiesbaden: Vieweg+Teubner. xi, 243 p.EUR 24.90 (2009). ISBN 978-3-8348-0381-8/pbk

This introduction to ordinary differential equations (ODEs) is modern in two aspects:1) It emphasizes the ‘dynamical systems’ point of view, i.e., the consideration of allsolutions of a given equation, their qualitative behavior and their geometric/topologicalstructure in phase space.2) Numerical exploration of ODEs is introduced along with the analytical theory (thiscorresponds to present-day methods of research).Both Maple and Matlab routines for numerical integration are discussed in two appen-dices. After classical introductory themes like existence and uniqueness, linear equa-tions, continuous and differentiable dependence, elementary solution methods, the au-thors present numerical schemes. The second part of the book treats dynamical topicslike (linearized and Liapunov) stability, Poincare-Bendixson-theory, attractors and bi-furcations, Hamiltonian systems. The last chapter gives interesting applications fromdifferent natural sciences. Each chapter concludes with a number of exercises, some ofwhich are numerical experiments.The relatively large number of topics has the logical consequence that most of themare touched somewhat briefly – quite natural for an introduction. A typical lecture onODEs will not exactly follow the book, but have considerable overlap with its contents.Thus it certainly provides a valuable companion for the students, also in view of theprice. It seems that the book is currently available in German language only.

Bernhard Lani-Wayda (Giessen)Keywords : introduction to ordinary differential equations; dynamical systems; numer-ical methods; applications in natural sciencesClassification :

∗34-01 Textbooks (ordinary differential equations)34Axx General theory ODE34Cxx Qualitative theory of solutions of ODE34Dxx Stability theory of ODE34-04 Machine computation, programs (ordinary differential equations)

Zbl 1159.93013

Grune, L.; Pannek, J.Practical NMPC suboptimality estimates along trajectories. (English)Syst. Control Lett. 58, No. 3, 161-168 (2009). ISSN 0167-6911http://dx.doi.org/10.1016/j.sysconle.2008.10.012http://www.sciencedirect.com/science/journal/01676911

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Summary: We develop and illustrate methods for estimating the degree of suboptimal-ity of receding horizon schemes with respect to infinite horizon optimal control. Theproposed a posteriori and a priori methods yield estimates which are evaluated onlinealong the computed closed-loop trajectories and only use numerical information whichis readily available in the scheme.

Keywords : model predictive control; suboptimality estimates; stabilityClassification :

∗93B40 Computational methods in systems theory93B51 Design techniques in systems theory49M30 Methods of successive approximation, not based on necessary cond.

Zbl pre05661857

Karafyllis, Iasson; Grune, LarsFeedback stabilization methods for the numerical solution of systems of or-dinary differential equations. (English)Simos, Theodore E. (ed.) et al., Numerical analysis and applied mathematics. Inter-national conference on numerical analysis and applied mathematics 2009, Rethymno,Crete, Greece, September 18–22, 2009. Vol. 1. Melville, NY: American Institute ofPhysics (AIP). AIP Conference Proceedings 1168, 1, 152-155 (2009). ISBN 978-0-7354-0705-3/hbk; ISBN 978-0-7354-0709-1/sethttp://dx.doi.org/10.1063/1.3241391

Keywords : nonlinear acoustics; numerical analysis; differential equationsClassification :

∗93-99 Systems and control

Zbl 1157.93028

Camilli, Fabio; Grune, Lars; Wirth, FabianControl Lyapunov functions and Zubov’s method. (English)SIAM J. Control Optim. 47, No. 1, 301-326 (2008). ISSN 0363-0129; ISSN 1095-7138http://dx.doi.org/10.1137/06065129Xhttp://epubs.siam.org/sam-bin/dbq/toclist/SICON

Summary: For finite-dimensional nonlinear control systems we study the relation be-tween asymptotic null-controllability and control Lyapunov functions. It is shown thatcontrol Lyapunov functions may be constructed on the domain of asymptotic null-controllability as viscosity solutions of a first order PDE that generalizes Zubov’s equa-tion. The solution is also given as the value function of an optimal control problemfrom which several regularity results may be obtained.

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Keywords : asymptotic null-controllability; control Lyapunov functions; Hamilton-Jacobi-Bellman equation; viscosity solutions; Zubov’s methodClassification :

∗93D10 Popov-type stability of feedback systems35B37 PDE in connection with control problems49L25 Viscosity solutions93B05 Controllability

Zbl 1147.49019

Grune, L.; Junge, O.Global optimal control of perturbed systems. (English)J. Optim. Theory Appl. 136, No. 3, 411-429 (2008). ISSN 0022-3239; ISSN 1573-2878http://dx.doi.org/10.1007/s10957-007-9312-zhttp://www.springerlink.com/openurl.asp?genre=journalissn=0022-3239

Summary: We propose a new numerical method for the computation of the optimalvalue function of perturbed control systems and associated globally stabilizing optimalfeedback controllers. The method is based on a set-oriented discretization of the statespace in combination with a new algorithm for the computation of shortest paths inweighted directed hypergraphs. Using the concept of multivalued game, we prove theconvergence of the scheme as the discretization parameter goes to zero.

Keywords : optimal control; dynamic games; set-oriented numerics; graph theoryClassification :

∗49M25 Finite difference methods93C73 Perturbations in control systems93D10 Popov-type stability of feedback systems91A25 Dynamic games

Zbl 1138.93350

Grune, Lars; Worthmann, Karl; Nesic, DraganContinuous-time controller redesign for digital implementation: a trajectorybased approach. (English)Automatica 44, No. 1, 225-232 (2008). ISSN 0005-1098http://dx.doi.org/10.1016/j.automatica.2007.05.003http://www.sciencedirect.com/science/journal/00051098

Summary: Given a continuous-time nonlinear closed loop system, we investigate sampled-data feedback laws for which the trajectories of the sampled-data closed loop systemconverge to the continuous-time trajectories with a prescribed rate of convergence as thelength of the sampling interval tends to zero. We derive necessary and sufficient condi-tions for the existence of such sampled-data feedback laws and – in case of existence –provide explicit redesign formulas and algorithms for these controllers.

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Keywords : nonlinear sampled-data control; convergence rate; Taylor series expansionClassification :

∗93B52 Feedback control93C57 Sampled-data control systems

Zbl pre05660320

Grune, Lars; Semmler, WilliAsset pricing with loss aversion. (English)J. Econ. Dyn. Control 32, No. 10, 3253-3274 (2008). ISSN 0165-1889http://dx.doi.org/10.1016/j.jedc.2008.01.002http://www.sciencedirect.com/science/journal/01651889

Summary: The use of standard preferences for asset pricing has not been very suc-cessful in matching asset price characteristics, such as the risk-free interest rate, equitypremium and the Sharpe ratio, to time series data. Behavioral finance has recently pro-posed more realistic preferences such as those with loss aversion. Research is startingto explore the implications of behaviorally founded preferences for asset price charac-teristics. Encouraged by some studies of S. Benartzi and R. H. Thaler [Q. J. Econ.110, No.1, 73–92 (1995; Zbl 0829.90040)] and N. Barberis, M. Huang and T. Santos[Q. J. Econ. 116, No.1, 1–53 (2001; Zbl 0979.91025)] we study asset pricing with lossaversion in a production economy. Here, we employ a stochastic growth model and usea stochastic version of a dynamic programming method with an adaptive grid scheme tocompute the above mentioned asset price characteristics of a model with loss aversion inpreferences. As our results show using loss aversion we get considerably better resultsthan one usually obtains from pure consumption-based asset pricing models includingthe habit formation variant.

Keywords : behavioral finance; loss aversion; stochastic growth models; asset pricingand stochastic dynamic programmingClassification :

∗91B2591B62 Dynamic economic models etc.91B70 Stochastic models90C39 Dynamic programming91G60

Zbl pre05533552

Grune, LarsOptimization based stabilization of nonlinear control systems. (English)Lirkov, Ivan (ed.) et al., Large-scale scientific computing. 6th international conference,LSSC 2007, Sozopol, Bulgaria, June 5–9, 2007. Revised papers. Berlin: Springer.Lecture Notes in Computer Science 4818, 52-65 (2008). ISBN 978-3-540-78825-6/pbkhttp://dx.doi.org/10.1007/978-3-540-78827-05

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Summary: We present a general framework for analysis and design of optimizationbased numerical feedback stabilization schemes utilizing ideas from relaxed dynamicprogramming. The application of the framework is illustrated for a set valued and graphtheoretic offline optimization algorithm and for receding horizon online optimization.

Classification :∗65-99 Numerical analysis

Zbl pre05375331

Grune, LarsInput-to-state stability, numerical dynamics and sampled-data control. (Eng-lish)GAMM-Mitt. 31, No. 1, 94-114 (2008). ISSN 0936-7195http://dx.doi.org/10.1002/gamm.200890005http://www.wiley-vch.de/publish/dt/journals/alphabeticIndex/2250/

Classification :∗93D25 Input-output approaches to stability of control systems93D09 Robust stability of control systems65L07 Numerical investigation of stability of solutions of ODE93C57 Sampled-data control systems

Zbl 1161.91447

Grune, Lars; Semmler, WilliAsset pricing with dynamic programming. (English)Comput. Econ. 29, No. 3-4, 233-265 (2007). ISSN 0927-7099; ISSN 1572-9974http://dx.doi.org/10.1007/s10614-006-9063-1http://www.springerlink.com/openurl.asp?genre=journalissn=0927-7099

Summary: The study of asset price characteristics of stochastic growth models suchas the risk-free interest rate, equity premium, and the Sharpe-ratio has been limitedby the lack of global and accurate methods to solve dynamic optimization models.In this paper, a stochastic version of a dynamic programming method with adaptivegrid scheme is applied to compute the asset price characteristics of a stochastic growthmodel. The stochastic growth model is of the type as developed by [Brock and Mirman(1972), Journal of Economic Theory, 4, 479-513 and Brock (1979), Part I: The growthmodel (pp. 165-190). New York: Academic Press; The economies of information anduncertainty (pp. 165-192). Chicago: University of Chicago Press. (1982). It hasbecome the baseline model in the stochastic dynamic general equilibrium literature.In a first step, in order to test our procedure, it is applied to this basic stochasticgrowth model for which the optimal consumption and asset prices can analytically becomputed. Since, as shown, our method produces only negligible errors, as compared tothe analytical solution, in a second step, we apply it to more elaborate stochastic growthmodels with adjustment costs and habit formation. In the latter model preferences are

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not time separable and past consumption acts as a constraint on current consumption.This model gives rise to an additional state variable. We here too apply our stochasticversion of a dynamic programming method with adaptive grid scheme to compute theabove mentioned asset price characteristics. We show that our method is very suitable tobe used as solution technique for such models with more complicated decision structure.

Keywords : Stochastic growth models; Asset pricing; Stochastic dynamic programming;Adaptive gridClassification :

∗91B62 Dynamic economic models etc.91B70 Stochastic models91B28 Finance etc.90C39 Dynamic programming90C15 Stochastic programming

Zbl 1128.37010

Grune, Lars; Kloeden, Peter E.; Siegmund, Stefan; Wirth, Fabian R.Lyapunov’s second method for nonautonomous differential equations. (Eng-lish)Discrete Contin. Dyn. Syst. 18, No. 2-3, 375-403 (2007). ISSN 1078-0947; ISSN1553-5231http://www.aimsciences.org/journals/dcdsA/online.jsp

Converse Lyapunov theorems are important, at least theoretically, for deducing exis-tence of Lyapunov functions from stability of invariant sets. The authors prove converseLyapunov theorems for the pullback, forward, and uniform attractors concentrating at-tention on obtaining Lyapunov functions that recover certain attraction rates in termsof given comparison functions from the classes K and KL. Recall that K contains allcontinuous functions γ : R+ → R+ such that γ(0) = 0 and γ is strictly increasing,whereas KL contains all continuous functions β : R2

+ → R+ that belong to class K inthe first argument and decrease monotonically to zero in the second argument. It isshown how the different notions of stability and attractivity can be characterized interms of attraction rates provided by comparison functions.First the notions of the pullback, forward, and uniform attractors are introduced withrespect to the attraction of arbitrary compact sets, which implies stability propertiestoo. Lyapunov functions are defined, and it is shown that if the base space of the skewproduct flow is compact, then only the maximal invariant set can posses Lyapunovfunctions. It is proved that a skew product flow satisfies a decay condition expressedin terms of comparison functions if and only if there exists a Lyapunov function thatcharacterizes this type of decay. Then it is demonstrated how different notions ofstability and attractivity may be equivalently expressed in terms of nonautonomouscomparison functions. Finally, Lyapunov and converse Lyapunov theorems for differentstability notions are established.

Yuri V. Rogovchenko (Kalmar)

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Keywords : stability; Lyapunov’s direct method; nonautonomous differential equations;converse Lyapunov theorems; skew product flows; attractorsClassification :

∗37B55 Nonautonomous dynamical systems34D20 Lyapunov stability of ODE37B25 Lyapunov functions and stability93D30 Scalar and vector Lyapunov functions

Zbl pre05199899

Becker, Stephanie; Grune, Lars; Semmler, WilliComparing accuracy of second-order approximation and dynamic program-ming. (English)Comput. Econ. 30, No. 1, 65-91 (2007). ISSN 0927-7099; ISSN 1572-9974http://dx.doi.org/10.1007/s10614-007-9087-1http://www.springerlink.com/openurl.asp?genre=journalissn=0927-7099

Summary: The accuracy of the solution of dynamic general equilibrium models hasbecome a major issue. Recent papers, in which second-order approximations have beensubstituted for first-order, indicate that this change may yield a significant improvementin accuracy. Second order approximations have been used with considerable successwhen solving for the decision variables in both small and large-scale models. Addition-ally, the issue of accuracy is relevant for the approximate solution of value functions.In numerous dynamic decision problems, welfare is usually computed via this same ap-proximation procedure. However, Kim and Kim ( Journal of International Economics,60, 471 - 500, 2003) have found a reversal of welfare ordering when they moved fromfirst- to second-order approximations. Other researchers, studying the impact of mon-etary and fiscal policy on welfare, have faced similar challenges with respect to theaccuracy of approximations of the value function. Employing a base-line stochasticgrowth model, this paper compares the accuracy of second-order approximations anddynamic programming solutions for both the decision variable and the value function aswell. We find that, in a neighborhood of the equilibrium, the second-order approxima-tion method performs satisfactorily; however, on larger regions, dynamic programmingperforms significantly better with respect to both the decision variable and the valuefunction.

Keywords : Dynamic general equilibrium model; Approximation methods; Second-orderapproximation; Dynamic programmingClassification :

∗91B62 Dynamic economic models etc.91B50 Equilibrium in economics90C39 Dynamic programming

Zbl 1135.93033

Camilli, Fabio; Cesaroni, Annalisa; Grune, Lars; Wirth, Fabian

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Stabilization of controlled diffusions and Zubov’s method. (English)Stoch. Dyn. 6, No. 3, 373-393 (2006). ISSN 0219-4937http://dx.doi.org/10.1142/S0219493706001803http://www.worldscinet.com/sd/sd.shtml

Summary: We consider a controlled stochastic system which is exponentially stabilizablein probability near an attractor. Our aim is to characterize the set of points whichcan be driven by a suitable control to the attractor with either positive probabilityor with probability one. This will be done by associating to the stochastic system asuitable control problem and the corresponding Zubov equation. We then show thatthis approach can be used as a basis for numerical computations of these sets.

Keywords : controlled diffusions; stochastic control systems; domain of null controlla-bility; control Lyapunov functions; viscosity solutions; Zubov’s methodClassification :

∗93E15 Stochastic stability93E20 Optimal stochastic control (systems)49L25 Viscosity solutions93B05 Controllability

Zbl 1100.93022

Grune, L.; Kloeden, P.E.Higher order numerical approximation of switching systems. (English)Syst. Control Lett. 55, No. 9, 746-754 (2006). ISSN 0167-6911http://dx.doi.org/10.1016/j.sysconle.2006.03.002http://www.sciencedirect.com/science/journal/01676911

Summary: Higher order numerical schemes for affine nonlinear control systems de-veloped elsewhere by the authors are adapted to switching systems with prescribedswitching times. In addition the calculation of the required multiple switching controlintegrals is discussed. These schemes are particularly useful in situations involving veryrapid switching.

Keywords : switching systems; affine control systems; Taylor numerical schemes; deriva-tive free schemes; commutative switching systems; multiple switching control integralsClassification :

∗93C10 Nonlinear control systems93B40 Computational methods in systems theory

Zbl 1100.93044

Grune, Lars; Saint-Pierre, PatrickAn invariance kernel representation of ISDS Lyapunov functions. (English)Syst. Control Lett. 55, No. 9, 736-745 (2006). ISSN 0167-6911

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http://dx.doi.org/10.1016/j.sysconle.2006.02.003http://www.sciencedirect.com/science/journal/01676911

Summary: We apply set valued analysis techniques in order to characterize the input-to-state dynamical stability (ISDS) property, a variant of the well known input-to-statestability (ISS) property. Using a suitable augmented differential inclusion we are able tocharacterize the epigraphs of minimal ISDS Lyapunov functions as invariance kernels.This characterization gives new insight into local ISDS properties and provides a basisfor a numerical approximation of ISDS and ISS Lyapunov functions via set orientednumerical methods.

Keywords : input-to-state stability; invariance kernel; Lyapunov functions; set valuedanalysis; set oriented numericsClassification :

∗93D30 Scalar and vector Lyapunov functions93D20 Asymptotic stability of control systems

Zbl 1100.93042

Nesic, Dragan; Grune, LarsA receding horizon control approach to sampled-data implementation ofcontinuous-time controllers. (English)Syst. Control Lett. 55, No. 8, 660-672 (2006). ISSN 0167-6911http://dx.doi.org/10.1016/j.sysconle.2005.09.013http://www.sciencedirect.com/science/journal/01676911

Summary: We propose a novel way for sampled-data implementation (with the zeroorder hold assumption) of continuous-time controllers for general nonlinear systems.We assume that a continuous-time controller has been designed so that the continuous-time closed-loop satisfies all performance requirements. Then, we use this control lawindirectly to compute numerically a sampled-data controller. Our approach exploitsa model predictive control (MPC) strategy that minimizes the mismatch between thesolutions of the sampled-data model and the continuous-time closed-loop model. Wepropose a control law and present conditions under which stability and sub-optimalityof the closed loop can be proved. We only consider the case of unconstrained MPC. Weshow that the recent results in [G. Grimm, M.J. Messina, A.R. Teel, S. Tuna, Modelpredictive control: for want of a local control Lyapunov function, all is not lost, IEEETrans. Automat. Control 2004, to appear] can be directly used for analysis of stabilityof our closed-loop system.

Keywords : controller design; stabilization; sampled-data; nonlinear; receding horizoncontrol; model predictive controlClassification :

∗93D21 Adaptive and robust stabilization93D30 Scalar and vector Lyapunov functions

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93C10 Nonlinear control systems

Zbl 1100.65056

Bauer, Florian; Grune, Lars; Semmler, WilliAdaptive spline interpolation for Hamilton-Jacobi-Bellman equations. (Eng-lish)Appl. Numer. Math. 56, No. 9, 1196-1210 (2006). ISSN 0168-9274http://dx.doi.org/10.1016/j.apnum.2006.03.011http://www.sciencedirect.com/science/journal/01689274

The authors study an adaptive discretization technique using cubic spline interpolationand the performance of adaptive spline interpolation in semi-Lagrangian discretizationschemes for Hamilton-Jacobi-Bellman (H-J-B) equations, respectively. The approachturns out to be efficient in case of smooth solutions which is investigated analyticallyand illustrated by a numerical example.Main result: For smooth solutions adaptive spline approximations perform very welland can considerably improve the results obtained with both adaptive low order andnonadaptive high order methods. For nonsmooth solutions (the authors always under-stand the solutions of the H-J-B equation in the viscosity solutions sense) numericalinstabilities may occur which can seriously affect and even destroy the convergence ofthe scheme. The local approximation properties of cubic splines on locally refined gridsby a theoretical analysis are investigated. How the proposed method performs in prac-tice is shown by the numerical examples. Finally the authors (using those examples)also illustrate numerical stability.

Jan Lovısek (Bratislava)Keywords : viscosity solution; optimal control; adaptive discretization; spline interpo-lation; adaptive grids; fixed point equation; numerical example; convergence; numericalstabilityClassification :

∗65K10 Optimization techniques (numerical methods)49L25 Viscosity solutions49M25 Finite difference methods

Zbl 1129.93500

Grune, Lars; Junge, OliverA set oriented approach to optimal feedback stabilization. (English)Syst. Control Lett. 54, No. 2, 169-180 (2005). ISSN 0167-6911http://dx.doi.org/10.1016/j.sysconle.2004.08.005http://www.sciencedirect.com/science/journal/01676911

Summary: We present a numerical construction of an optimal control based feedbacklaw for the stabilization of discrete time nonlinear control systems. The feedback is

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based on a recently developed numerical solution method for optimal control problemsusing set oriented and graph theoretic algorithms. We show how this method can beused to construct approximately optimal and stabilizing feedback laws and present ana posteriori error estimation technique for the adaptive generation of the underlying setoriented space discretization.

Keywords : feedback stabilization; nonlinear control system; optimal control; set ori-ented numerics; shortest pathClassification :

∗93D15 Stabilization of systems by feedback49N35 Closed-loop controls93B40 Computational methods in systems theory

Zbl 1115.93064

Nesic, Dragan; Grune, LarsLyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation. (English)Automatica 41, No. 7, 1143-1156 (2005). ISSN 0005-1098http://dx.doi.org/10.1016/j.automatica.2005.03.001http://www.sciencedirect.com/science/journal/00051098

Nonlinear sampled data systems are considered in this paper. It is assumed that thecontinuous plant can be described by state space description linear in the control inputu. A control law for the continuous system with an appropriate Lyapunov functionmust be available.The main goal of the paper is to adapt the continuous-time controller for sampled datasystems with zero order hold.For this purpose the continuous time control law is replaced by a power series in thesampling time T with the first term identical to the continuous time controller. Theother terms are determined by using a Fliess series expansion of the Lyapunov differenefor the discrete time system. The Lyapunov difference is based on the known Lyapunovfunction used for the continuous-time controller design.A procedure for developing the discrete-time controller is described. The proposedmethod is illustrated by two numerical examples.

Rudolf Tracht (Essen)Keywords : nonlinear sampled data systems; controller design; Lyapunov functionClassification :

∗93C57 Sampled-data control systems93B35 Sensitivity (robustness) of control systems93D09 Robust stability of control systems

Zbl 1107.91361

Grune, Lars; Semmler, Willi; Sieveking, Malte

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Creditworthiness and thresholds in a credit market model with multipleequilibria. (English)Econ. Theory 25, No. 2, 287-315 (2005). ISSN 0938-2259; ISSN 1432-0479http://dx.doi.org/10.1007/s00199-003-0442-8http://www.springerlink.com/content/1432-0479/

Summary: The paper studies creditworthiness in a model with endogenous credit costand debt constraints. Such a model can give rise to multiple candidates for steadystate equilibria. We use new analytical techniques such as dynamic programming (DP)with flexible grid size to find solutions and to locate thresholds that separate differentdomains of attraction. More specifically, we (1) compute present value borrowing con-straints and thus creditworthiness, (2) locate thresholds where the dynamics separate todifferent domains of attraction, (3) show jumps in the decision variable, (4) distinguishbetween optimal and non-optimal steady states, (5) demonstrate how creditworthinessand thresholds change with change of the credit cost function of the debtor and (6)explore the impact of debt ceilings and consumption paths on creditworthiness.

Keywords : creditworthiness; default risk; imperfect capital markets; multiple equilibria;asset pricing; dynamic programming.Classification :

∗91B62 Dynamic economic models etc.49L20 Dynamic programming method (infinite-dimensional problems)90C39 Dynamic programming

Zbl 1078.93060

Grune, LarsQuantitative aspects of the input-to-state-stability property. (English)de Queiroz, Marcio (ed.) et al., Optimal control, stabilization and nonsmooth analy-sis. Papers from the Louisiana conference on mathematical control theory (MCT’03),Louisiana State University, Baton Rouge, LA, USA, April 10–13, 2003. Berlin: Springer.Lecture Notes in Control and Information Sciences 301, 215-230 (2004). ISBN 3-540-21330-9/pbk

The author introduces the concept of input-to-state dynamical stability (ISDS), a gener-alization of Sontag’s input-to-state stability (ISS) property. One of the most importantfeatures of the ISS property is that it can be characterized by a dissipation inequalityusing an ISS Lyapunov function. The ISDS property also admits an ISDS Lyapunovfunction that characterizes the ISDS qualitatively and also permits estimation of decayrate, overshoot gain and robustness gain.The setting consists of nonlinear systems of the form x(t) = f(x(t), u(t)) where f :Rn × Rm → Rn is continuous, and for two compact subsets K ⊂ Rn and W ⊂ Rm,there is a constant L = L(K, W ) such that ‖f(x, u) − f(y, u)‖ ≤ L‖x − y‖ for allx, y ∈ K and u ∈ W . The perturbation function u is assumed to be measurable andlocally essentially bounded.

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The author’s primary results are two theorems. The first shows how to pass from the ISSto the ISDS formulation. The second provides a characterization of the ISDS propertyvia Lyapunov functions.Applications are presented in numerical anaysis and in control theory. The numericalanalysis example involves the computation of attractors, and specifically the rates ofconvergence. The control theory applications include estimation of a nonlinear stabilitymargin as well as analysis of stability of coupled systems, with estimates of overshootand decay rates.

William J. Satzer jun. (St. Paul, Minnesota)Keywords : input-to-state stability; input-to-state dynamic stability; nonlinear stabilitytheory; Lyapunov functionsClassification :

∗93D21 Adaptive and robust stabilization

Zbl 1074.65009

Grune, LarsError estimation and adaptive discretization for the discrete stochastic Hamilton-Jacobi-Bellman equation. (English)Numer. Math. 99, No. 1, 85-112 (2004). ISSN 0029-599X; ISSN 0945-3245http://dx.doi.org/10.1007/s00211-004-0555-4http://link.springer.de/link/service/journals/00211/

The dynamic programming method is a well known technique for the numerical so-lution of optimal control problems. Generalizing the technique and results from thedeterministic case [cf. the author, ibid. 75, 319–337 (1997; Zbl 0880.65045)], the au-thor obtains a posteriori error estimates for the space discretization of the stochasticHamilton-Jacobi-Bellman equation. This method gives full global information aboutthe optimal value function of the related stochastic optimal control problem. Thereforea feedback optimal control can be obtained.It is also demonstrated that the a posteriori error estimates are efficient and reliable forthe numerical approximation of PDEs and they allow to derive a bound for the numericalerror corresponding to the derivatives. The asymptotic behavior of the error estimateswith respect to the size of the grid elements is also investigated. Finally, an adaptivespace discretization scheme is developed and numerical examples are presented.

Viorel Arnautu (Iasi)Keywords : stochastic optimal control; stochastic Hamilton-Jacobi-Bellman equation; aposteriori error estimates; feedback optimal control; numerical examplesClassification :

∗65C30 Stochastic differential and integral equations60H15 Stochastic partial differential equations60H35 Computational methods for stochastic equations49J55 Optimal stochastic control (existence)65K10 Optimization techniques (numerical methods)49M25 Finite difference methods

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65N15 Error bounds (BVP of PDE)49L20 Dynamic programming method (infinite-dimensional problems)

Zbl 1051.93015

Grune, LarsRobust asymptotic controllability under time-varying perturbations. (Eng-lish)Stoch. Dyn. 4, No. 3, 297-316 (2004). ISSN 0219-4937http://dx.doi.org/10.1142/S0219493704001085http://www.worldscinet.com/sd/sd.shtml

A perturbed control system of the form x(t) = f(x(t), u(t), w(t)) is considered, wheref : Rd × U × W → Rd is continuous and globally Lipschitz in x uniformly for all uand w, with control functions u ∈ U := u : R → U ⊂ Rn, u measurable and withtime-varying perturbations w ∈ W = w ∈ R → W ⊂ Rl, w measurable. The propertyof asymptotic controllability of compact sets is determined.The effects of discretization in both time and space are studied and suitable classesof perturbations for modelling these effects are identified. The action of such classesof time-varying perturbations on asymptotically controllable sets is investigated. Ap-propriate robustness properties are introduced and it is proven that there are inherentproperties for asymptotically controllable sets under these classes of perturbations.

Victor Sharapov (Volgograd)Keywords : robust controllability; time-varying perturbation; discretizationClassification :

∗93B05 Controllability93B35 Sensitivity (robustness) of control systems

Zbl pre05362848

Grune, Lars; Semmler, WilliUsing dynamic programming with adaptive grid scheme for optimal controlproblems in economics. (English)J. Econ. Dyn. Control 28, No. 12, 2427-2456 (2004). ISSN 0165-1889http://dx.doi.org/10.1016/j.jedc.2003.11.002http://www.sciencedirect.com/science/journal/01651889

Summary: The study of the solutions of dynamic models with optimizing agents hasoften been limited by a lack of available analytical techniques to explicitly find theglobal solution paths. On the other hand, the application of numerical techniques suchas dynamic programming to find the solution in interesting regions of the state wasrestricted by the use of fixed grid size techniques. Following Grune (Numer. Math.75 (3) (1997) 319; University of Bayreuth, submitted, 2003), in this paper an adaptivegrid scheme is used for finding the global solutions of discrete time Hamilton-Jacobi-

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Bellman equations. Local error estimates are established and an adapting iteration forthe discretization of the state space is developed. The advantage of the use of adaptivegrid scheme is demonstrated by computing the solutions of one- and two-dimensionaleconomic models which exhibit steep curvature, complicated dynamics due to multipleequilibria, thresholds (Skiba sets) separating domains of attraction and periodic solu-tions. We consider deterministic and stochastic model variants. The studied examplesare from economic growth, investment theory, environmental and resource economics.

Keywords : dynamic optimization; dynamic programming; adaptive grid schemeClassification :

∗91-99 Social and behavioral sciences93-99 Systems and control

Zbl 1123.60311

Camilli, Fabio; Grune, LarsCharacterizing attraction probabilities via the stochastic Zubov equation.(English)Discrete Contin. Dyn. Syst., Ser. B 3, No. 3, 457-468 (2003). ISSN 1531-3492; ISSN1553-524Xhttp://dx.doi.org/10.3934/dcdsb.2003.3.457http://www.aimsciences.org/journals/dcdsB/dcdsbonline.jsp

Summary: A stochastic differential equation with an a.s. locally stable compact set isconsidered. The attraction probabilities to the set are characterized by the sublevel setsof the limit of a sequence of solutions to 2nd order partial differential equations. Twonumerical examples illustrating the method are presented.

Keywords : stochastic differential equation; almost sure exponential stability; Zubov’smethod; viscosity solutionClassification :

∗60H10 Stochastic ordinary differential equations93E15 Stochastic stability49L25 Viscosity solutions34F05 ODE with randomness

Zbl 1058.65081

Grune, LarsAttraction rates, robustness, and discretization of attractors. (English)SIAM J. Numer. Anal. 41, No. 6, 2096-2113 (2003). ISSN 0036-1429; ISSN 1095-7170http://dx.doi.org/10.1137/S003614290139411Xhttp://epubs.siam.org/sam-bin/dbq/toclist/SINUM

Summary: We investigate necessary and sufficient conditions for the convergence ofattractors of discrete time dynamical systems induced by numerical one-step approxi-

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mations of ordinary differential equations (ODEs) to an attractor for the approximatedODE. We show that both the existence of uniform attraction rates (i.e., uniform speedof convergence toward the attractors) and uniform robustness with respect to pertur-bations of the numerical attractors are necessary and sufficient for this convergenceproperty. In addition, we can conclude estimates for the rate of convergence in theHausdorff metric.

Keywords : numerical one-step approximation; attractor; attraction rate; robustness;convergence; discrete time dynamical systemsClassification :

∗65L20 Stability of numerical methods for ODE65L06 Multistep, Runge-Kutta, and extrapolation methods34D45 Attractors34E10 Asymptotic perturbations (ODE)37C30 Functional analytic techniques in dynamical systems37D45 Strange attractors, chaotic dynamics65P40 Nonlinear stabilities

Zbl 1044.93053

Grune, Lars; Nesic, DraganOptimization-based stabilization of sampled-data nonlinear systems via theirapproximate discrete-time models. (English)SIAM J. Control Optimization 42, No. 1, 98-122 (2003). ISSN 0363-0129; ISSN 1095-7138http://dx.doi.org/10.1137/S036301290240258Xhttp://epubs.siam.org/sam-bin/dbq/toclist/SICON

Assume that an input system

(1) x = f(x, u), u ∈ U

and a sampling time T > 0 are given. If the system equations can be integrated for anyconstant u ∈ U and initial state x0, and assuming that the solution φ(t, x0, u) exists onthe whole interval [0, T ], one can associate to (1) the exact discrete-time model

(2) xk+1 = F eT (xk, uk),

where F eT (x, u) = φ(T, x, u). Since in general the system equations cannot be explicitly

integrated, some numerical algorithm has to be used. This is the same as replacing (2)by an approximate discrete-time system

(3) xk+1 = F aT,h(xk, uk),

where h is a possible new modelling parameter. Control design is often based on theapproximate model (3). However, examples show that control laws which are stabilizingfor (3), may fail when applied to the exact model (2).In this paper, the authors consider stabilizing control laws obtained by solving cer-tain optimization problems (on the finite or infinite horizon) based on the approximatemodel, and give conditions which ensure that the same control law stabilizes the exact

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model as well. The results cover a variety of situations (for instance, the case whereT = h and the case where T and h are independent).

Andrea Bacciotti (Torino)Keywords : controller design; asymptotic controllability; stabilization; numerical meth-ods; optimal control; approximate model; discretization; samplingClassification :

∗93D15 Stabilization of systems by feedback93C57 Sampled-data control systems93C10 Nonlinear control systems65P40 Nonlinear stabilities49N35 Closed-loop controls93C55 Discrete-time control systems93B52 Feedback control

Zbl 1024.93023

Grune, Lars; Kloeden, Peter E.Numerical schemes of higher order for a class of nonlinear control systems.(English)Dimov, Ivan (ed.) et al., Numerical methods and applications. 5th international con-ference, NMA 2002, Borovets, Bulgaria, August 20-24, 2002. Revised papers. Berlin:Springer. Lect. Notes Comput. Sci. 2542, 213-220 (2003). ISBN 3-540-00608-7/pbkhttp://link.springer.de/link/service/series/0558/bibs/2542/25420213.htm

Summary: We extend a systematic method for the derivation of high order schemes foraffinely controlled nonlinear systems to a larger class of systems in which the controlvariables are allowed to appear nonlinearly in multiplicative terms. Using an adaptationof the stochastic Taylor expansion to control systems we construct Taylor schemes ofarbitrary high order and indicate how derivative free Runge-Kutta type schemes can beobtained.

Classification :∗93C10 Nonlinear control systems93B40 Computational methods in systems theory

Zbl 0994.00036

Colonius, Fritz (ed.); Grune, Lars (ed.)Dynamics, bifurcations, and control. 3rd nonlinear control workshop, KlosterIrsee, Germany, April 1–3, 2001. (English)Lecture Notes in Control and Information Sciences. 273. Berlin: Springer. xi, 303 p.EUR 82.00 (net); sFr 136.00; £52.00; $ 89.80 (2002). ISBN 3-540-42890-9

The articles of this volume will be reviewed individually.

Keywords : Irsee (Germany); Workshop; Proceedings; Nonlinear control workshop; Dy-namics; Bifurcations; Control

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Classification :∗00B25 Proceedings of conferences of miscellaneous specific interest93-06 Proceedings of conferences (systems and control)

Zbl 0991.37001

Grune, LarsAsymptotic behavior of dynamical and control systems under perturbationand discretization. (English)Lecture Notes in Mathematics. 1783. Berlin: Springer. ix, 231 p. EUR 32.95/net; sFr.55.00; £23.00; $ 45.80 (2002). ISBN 3-540-43391-0/pbkhttp://link.springer.de/link/service/series/0304/tocs/t1783.htm

This monograph (based on the author’s habilitation thesis) contributes to the numer-ical analysis of dynamical systems and control systems. It lies in the intersection ofperturbation theory of dynamical systems, numerical dynamics, control theory and(dynamic) game theory. Concepts and methods of all these areas are interwoven. Thegeneral theme is the behavior of attracting sets and attractors under discretization.This contribution is not, as many others, based on a priori structural assumptions likehyperbolicity. This assumption is already violated for most relevant dynamical sys-tems from applications; if controls or perturbations are present, they are restricted tosmall amplitudes. Thus, although it is possible (and also interesting) to develop thosetheories, their relevance, however, is very restricted.Grune uses the classical method of comparison functions for stability analysis, goingback to W. Hahns work. They allow for a quantitative description of stability proper-ties, and in the last few years, they have witnessed a renaissance in control theory. Heconsiders two main classes of problems: systems with internal disturbances and con-trolled systems. In order to understand the behavior of numerical approximations tothese systems, further (external) disturbances are introduced, which are of two differenttypes: for systems with internal disturbances, Grune takes time dependent disturbances;for control systems he considers external disturbances, generated by nonanticipatorystrategies, a concept from game theory. In the first case, so-called strong concepts areobtained, in the second case, weak concepts. Nonanticipatory perturbations make itpossible to model numerical errors for controlled systems.After the introduction and the preparatory Chapter 2 (mainly on the considered systemsclasses), Chapters 3 and 4 study attracting sets. Here the new concept of dynamicalInput-to-State-Stability is central. In Chapter 5 the connection between time and spacedicretization on the one hand and disturbed systems on the other is established. Thusthe theory developed in Chapters 3 and 4 becomes applicable to the analysis of nu-merical methods. This is performed in detail in Chapter 6 for attracting sets and forattractors. In Chapter 7 domains of attraction and reachable sets are discussed. Thetext is completed by 3 appendices: on viscosity solutions, on comparison functions andon numerical examples; finally, it contains a list of notation, a subject index and 129references.The monograph contains many new insights and results which cannot be reported here.

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An important complement is Appendix C which illustrates the main results by some –very successful – numerical experiments. This monograph lays foundations for a sys-tematic treatment of the numerical analysis of control systems with nonlinear dynamics;this is achieved by embedding this theory into numerical stability theory of dynamicalsystems. In my opinion this monograph will be a milestone in our understanding of nu-merical methods for nonlinear control systems and of perturbed systems and promisesto be very influential for the further work in this area.

Fritz Colonius (Augsburg)Keywords : dynamical systems; control systems; discretizationClassification :

∗37-02 Research exposition (Dynamical systems and ergodic theory)93-02 Research monographs (systems and control)65P40 Nonlinear stabilities37B25 Lyapunov functions and stability93D30 Scalar and vector Lyapunov functions93B05 Controllability93B35 Sensitivity (robustness) of control systems49L25 Viscosity solutions

Zbl 1006.93061

Camilli, Fabio; Grune, Lars; Wirth, FabianA generalization of Zubov’s method to perturbed systems. (English)SIAM J. Control Optimization 40, No.2, 496-515 (2001). ISSN 0363-0129; ISSN 1095-7138http://dx.doi.org/10.1137/S036301299936316Xhttp://epubs.siam.org/sam-bin/pii.pl?pii=S0363-0129-99-36316-Xhttp://epubs.siam.org/sam-bin/dbq/toclist/SICON

The authors consider the development of Zubov’s method for the determination of thedomain of attraction of an asymptotically stable fixed point. Zubov’s idea to reduce theproblem to the existence of a solution to a suitable partial differential equation for theLyapunov function is modified.

Yuri N.Sankin (Ul’yanovsk)Keywords : asymptotic stability; Zubov’s method; robust stability; domain of attraction;viscosity solutionsClassification :

∗93D20 Asymptotic stability of control systems93D30 Scalar and vector Lyapunov functions93D09 Robust stability of control systems49L25 Viscosity solutions

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Zbl 0998.65010

Grune, L.; Kloeden, P.E.Pathwise approximation of random ordinary differential equations. (English)BIT 41, No.4, 711-721 (2001). ISSN 0006-3835; ISSN 1572-9125http://dx.doi.org/10.1023/A:1021995918864http://www.springerlink.com/openurl.asp?genre=journalissn=0006-3835

This paper presents an averaging procedure for improving the Euler and Heun methodsof numerical approximation of the solution of the vector random ordinary differentialequation

dx

dt= G(t, ω) + g(t, ω)H(x, ω),

where G, g are Holder continuous in t, and H is once (for Euler) or twice (for Heun)continuously differentiable in x. It is shown that by averaging G and g over N equallyspaced values of t in each discretization subinterval, order of convergence 1 for the Eulermethod and 2 for the Heun method can be attained, whereas the standard Euler andHeun methods may have only fractional order. Two examples are given which demon-strate that the averaging procedure approximations can show a substantial reductionin error over the approximations generated by the standard Euler and Heun methods.

Melvin D.Lax (Long Beach)Keywords : Euler method; averaging method; error reduction; Heun methods; vectorrandom ordinary differential equation; convergenceClassification :

∗65C30 Stochastic differential and integral equations65L06 Multistep, Runge-Kutta, and extrapolation methods34F05 ODE with randomness65L20 Stability of numerical methods for ODE60H10 Stochastic ordinary differential equations60H35 Computational methods for stochastic equations65L70 Error bounds (numerical methods for ODE)

Zbl 0996.37013

Grune, L.; Kloeden, P.E.Discretization, inflation and perturbation of attractors. (English)Fiedler, Bernold (ed.), Ergodic theory, analysis, and efficient simulation of dynamicalsystems. Berlin: Springer. 399-416 (2001). ISBN 3-540-41290-5/hbk

The paper studies attractor properties in autonomous and nonautonomous systemsof differential equations using general properties of Lyapunov functions. The nonau-tonomous case is analyzed with respect to pullback convergence (for t0 → −∞, wheret0 is the current initial moment). Pullback attractors are studied.

Vladimir Rasvan (Craiova)

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Keywords : pullback attractors; nonautonomous systems; Lyapunov functions; pullbackconvergenceClassification :

∗37B25 Lyapunov functions and stability37M99 Approximation methods and numerical treatment of dynamical systems

Zbl 0992.65075

Grune, L.; Kloeden, P.E.Higher order numerical schemes for affinely controlled nonlinear systems.(English)Numer. Math. 89, No.4, 669-690 (2001). ISSN 0029-599X; ISSN 0945-3245http://dx.doi.org/10.1007/s002110000279http://link.springer.de/link/service/journals/00211/

Numerical schemes for affinely controlled systems received a special interest becausecomplex nonlinear control systems do not allow, in general, an analytic solution, thusrequiring numerical treatment for both analyses and controller design. The presentpaper develops a systematic method for the derivation of high-order schemes for affinelycontrolled nonlinear systems. The proposed stochastic Taylor expansion for controlsystems is essentially the same as the Fliess expansion that is well-known in controltheory, with the main difference adapted from stochastic calculus, which allows, inparticular, a transparent representation of the remainder term and a straightforwardderivation of approximations of an arbitrary desired order.The paper shows how derivative-free schemes can be obtained from numerical Taylorschemes of arbitrary order. This provides an efficient device for the construction of theright kind of Runge-Kutta schemes for the affinely controlled nonlinear systems. Severalsimplifications to the Taylor schemes based on a special additive or commutative controlstructure of the system are also discussed. Furthermore, the approximation of themultiple control integrals appearing in the proposed schemes is analyzed; in particular,the approximation by averaging for a single control function, and the approximation ofthe set of multiple control integrals for all measurable control functions. A numericalexample testing the Euler and Heun schemes for a 2-dimensional system with a singlecontrol illustrates the proposed method.

Silvia Curteanu (Iasi)Keywords : affinely controlled nonlinear systems; stochastic Taylor expansion; Fliessexpansion; numerical Taylor schemes; Runge-Kutta schemes; derivative-free schemes;numerical exampleClassification :

∗65K10 Optimization techniques (numerical methods)65L05 Initial value problems for ODE (numerical methods)93B40 Computational methods in systems theory93C15 Control systems governed by ODE

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Zbl 0991.65006

Cyganowski, S.; Grune, L.; Kloeden, P.E.Maple for stochastic differential equations. (English)Blowey, James F. (ed.) et al., Theory and numerics of differential equations. 9thEPSRC numerical analysis summer school, Univ. of Durham, GB, July 10-21, 2000.Berlin: Springer. Universitext. 127-177 (2001). ISBN 3-540-41846-6

Summary: This chapter introduces the MAPLE software package stochastic consistingof MAPLE routines for stochastic calculus and stochastic differential equations andfor constructing basic numerical methods for specific stochastic differential equations,with simple examples illustrating the use of the routines. A website address is givenfrom which the software can be downloaded and where up to late information aboutinstallment, new developments and literature can be found.

Keywords : MAPLE software package; stochastic calculus; stochastic differential equa-tionsClassification :

∗65C30 Stochastic differential and integral equations60H07 Stochastic calculus of variations and the Malliavin calculus60H10 Stochastic ordinary differential equations60H35 Computational methods for stochastic equations65Y15 Packaged methods68W30 Symbolic computation and algebraic computation

Zbl 0989.65144

Grune, LarsPersistence of attractors for one-step discretization of ordinary differentialequations. (English)IMA J. Numer. Anal. 21, No.3, 751-767 (2001). ISSN 0272-4979; ISSN 1464-3642http://dx.doi.org/10.1093/imanum/21.3.751http://imajna.oxfordjournals.org/

This paper deals with the behaviour of one step discretization methods with respect tothe attractors of the original differential system. A basic result in this direction wasproved by P. E. Kloeden and J. Lorenz [SIAM J. Numer. Anal 23, 986-995 (1986; Zbl0613.65083)]. who showed that if the original differential equations have an attractorthen the discretization possess absorbing sets which converge to the attractor as thetime step tends to zero. However the existence of numerical attractors does not implythe existence of a nearby attractor for the original differential system.The author studies this converse implication and proves that under some additionalassumption it holds true. More precisely the upper limit of the numerical attractors isan attractor of the original differential equation if and only if the numerical one stepscheme admits attracting sets with uniform attraction rate approximating this upperlimit set. Furthermore if the numerical attractors themselves are attracting with uniform

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rate, then they converge to some set A if and only if this set A is an attractor of theoriginal ordinary differential equation and an estimate of the rate of convergence maybe derived.

Manuel Calvo (Zaragoza)Keywords : one step methods; conservation of attractors; numerical attractors; conver-genceClassification :

∗65P40 Nonlinear stabilities65L06 Multistep, Runge-Kutta, and extrapolation methods37C70 Attractors and repellers, topological structure37M15 Symplectic integrators

Zbl 0988.65088

Grune, LarsAdaptive grid generation for evolutive Hamilton-Jacobi-Bellman equations.(English)Falcone, Maurizio (ed.) et al., Numerical methods for viscosity solutions and applica-tions. Singapore: World Scientific. Ser. Adv. Math. Appl. Sci. 59, 153-172 (2001).ISBN 981-02-4717-6

Summary: We present an adaptive grid generation for a class of evolutive Hamilton-Jacobi-Bellman equations. Using a two step (semi-Lagrangian) discretization of theunderlying optimal control problem we define a posteriori local error estimates for thediscretization error in space. Based on these estimates we present an iterative procedurefor the generation of adaptive grids and discuss implementational details for a suitablehierarchical data structure.

Keywords : semi-Lagrangian discretization; evolutive Hamilton-Jacobi-Bellman equa-tions; optimal control; error estimates; iterative procedure; adaptive gridsClassification :

∗65M50 Mesh generation and refinement (IVP of PDE)35L60 First-order nonlinear hyperbolic equations49L25 Viscosity solutions65M20 Method of lines (IVP of PDE)65M15 Error bounds (IVP of PDE)

Zbl 1020.93500

Colonius, Fritz; Kliemann, Wolfgang (Grune, Lars)The dynamics of control. With an appendix by Lars Grune. (English)Systems and Control: Foundations and Applications. Boston: Birkhauser. xii, 629 p.(2000). ISBN 0-8176-3683-8

This book is a treatise bringing together control theory and dynamical systems. Thecontrols are also interpreted as perturbations without however going so far as to provid-ing a simultaneous study of both controls seen as inputs and additional (time-varying)

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perturbations. In this new frame, the usual notions of controllability and stability areextended and investigated. The authors write that this is a “report about ongoingresearch” and not “a presentation of a complete history”. They point out some openareas: the cases of loss of accessibility, discrete cases, computational aspects, large scalesystems, bifurcation, stochastic aspects (including the problem of convergence of invari-ant measures for approximating flows to the nominal one), stabilization issues relatedto specific trajectories. A general nonlinear theory is presented in the first part, whilethe next one deals with linearization. The final part dicusses applications.Chapter two is of an introductory nature, showing how to express controlled ODEsas dynamical systems, explaining concepts of stability, leading via linearization to theLyapunov spectrum. Chapter three explains control sets and the notion of chain con-trollability (one sends with the controls from one point arbitrarily close to the next ina given set of points in the state space). Chapter four deals with the objects of the pre-vious chapter using an asymptotic approach. Control sets are related to mixing, whilechain control sets involve transitive bahaviour. Then the location of ω-limit sets of con-trolled trajectories are considered. They intersect the control sets and are contained inchain control sets, but the issue is to know if they are contained in the interior of such aset. Chapter five studies linear flows on vector bundles generalizing linear time-varyingsystems. The authors look at the Morse spectrum, its boundary points and relatedinvariant manifolds. Chapter six considers bilinear systems (Morse spectrum, Floquetspectrum, Lyapunov spectrum), and the results are specialized subsequently in the caseof a linearization at a singular point.Part three starts next: The one-dimensional case allows explicit computations. Thereader finds various practical examples and explorations including numerical ones on(robust) stability and stabilization. A chapter deals with the computation of the spec-trum in dimension two. There are four appendices which are useful. The first oneintroduces differential geometric tools (assuming that the dynamics involving state andcontrol is a direct product, a special case – it would have been a good idea to specifythe general configuration of relevance here since the authors are preoccupied with globalbehaviour); then the authors deal with dynamical systems basics (Morse decomposition,chains). Finally the focus is on numerical computations (orbits, spectrum (this last part,written by L. Grune, involves an optimal control method)). The bibliography contains333 entries, and an index is included.The book is very interesting. The dynamical systems perspective provides a welcomenew impetus to control theory, confirming once more the fertility of this discipline.

A.Akutowicz (Berlin)Keywords : dynamical systems; perturbations; controllability; stability; linearization;Lyapunov spectrum; chain controllability; mixing; ω-limit sets; linear flows; Morse spec-trum; Floquet spectrum; numerical computations; acessibility; control setsClassification :

∗93-02 Research monographs (systems and control)93C15 Control systems governed by ODE93B18 Linearizability of systems93D09 Robust stability of control systems93C73 Perturbations in control systems93B05 Controllability

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37N35 Dynamical systems in control37C60 Nonautonomous smooth dynamical systems34D08 Lyapunov exponents

Zbl 0988.65077

Camilli, Fabio; Grune, LarsNumerical approximation of the maximal solutions for a class of degenerateHamilton-Jacobi equations. (English)SIAM J. Numer. Anal. 38, No.5, 1540-1560 (2000). ISSN 0036-1429; ISSN 1095-7170http://dx.doi.org/10.1137/S003614299834798Xhttp://epubs.siam.org/sam-bin/pii.pl?pii=S003614299834798Xhttp://epubs.siam.org/sam-bin/dbq/toclist/SINUM

The authors propose an approximation scheme for a class of Hamilton-Jacobi problemsfor which uniqueness of the viscosity solution does not hold. This class includes theeikonal equation arising in the shape-from-shading problem. The scheme is based on atwo step discretization of the control problem associated with the regularized problem.A numerical estimate of the discretization error is also given.

T.C.Mohan (Chennai)Keywords : singular Hamilton-Jacobi equations; maximal solution; regularization; nu-merical approximation; degenerate Hamilton-Jacobi equations; error bound; viscositysolution; eikonal equation; shape-from-shading problemClassification :

∗65M06 Finite difference methods (IVP of PDE)49L25 Viscosity solutions35L60 First-order nonlinear hyperbolic equations65M15 Error bounds (IVP of PDE)65K10 Optimization techniques (numerical methods)

Zbl 0983.37032

Grune, LarsA uniform exponential spectrum for linear flows on vector bundles. (English)J. Dyn. Differ. Equations 12, No.2, 435-448 (2000). ISSN 1040-7294; ISSN 1572-9222http://dx.doi.org/10.1023/A:1009024610394http://www.springerlink.com/openurl.asp?genre=journalissn=1040-7294

It is known that spectral concepts for linear flows ϕ : R × E → E on vector bundlesπ : E → S with compact base space S can roughly be divided into two classes: one usingexponential growth rates and the other using topological characterizations of the flowprojected onto the projective bundle. Here the author introduces a spectral conceptthat lies somewhat in between these approaches. He considers the set of all possibleexponential growth rates in some finite time T > 0 with initial values in this set and

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defines the spectrum to consist of all accumulation points as T → ∞. He proves thatfor a connected compact invariant set this spectrum is a closed interval whose boundarypoints are Lyapunov exponents.

Messoud Efendiev (Berlin)Keywords : linear flow; uniform; exponential spectrum; Lyapunov exponent; accumula-tion pointClassification :

∗37D25 Nonuniformly hyperbolic systems34D08 Lyapunov exponents37C35 Orbit growth

Zbl 0974.49002

Grune, Lars; Wirth, FabianOn the rate of convergence of infinite horizon discounted optimal value func-tions. (English)Nonlinear Anal., Real World Appl. 1, No.4, 499-515 (2000). ISSN 1468-1218http://dx.doi.org/10.1016/S0362-546X(99)00288-6http://www.sciencedirect.com/science/journal/14681218http://www.sciencedirect.com/science/journal/14681218

The authors investigate the rate of convergence of the optimal value function of aninfinite horizon discounted optimal control problem as the discount rate tends to zero.Using the integration theorem for Laplace transforms the authors provide conditionson averaged functionals along suitable trajectories yielding quadratic pointwise conver-gence. From that the authors derive under appropriate controllability conditions criteriafor linear uniform convergence of the value functions and control sets. Applications ofthose results are given and an example is discussed in which both linear and slower ratesof convergence occur depending on the cost functional.

P.Neittaanmaki (Jyvaskyla)Keywords : rate of convergence; optimal value function; infinite horizon discountedoptimal control problemClassification :

∗49J15 Optimal control problems with ODE (existence)49K15 Optimal control problems with ODE (nec./ suff.)49M05 Methods of successive approximation based on necessary conditions

Zbl 0966.34045

Grune, Lars; Sontag, Eduardo D.; Wirth, FabianOn equivalence of exponential and asymptotic stability under changes ofvariables. (English)Fiedler, B. (ed.) et al., International conference on differential equations. Proceedingsof the conference, Equadiff ’99, Berlin, Germany, August 1-7, 1999. Vol. 2. Singapore:World Scientific. 850-852 (2000). ISBN 981-02-4989-6/hbk

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Summary: The authors show that uniformly global asymptotic stability for a familyof ordinary differential equations is equivalent to uniformly global exponential stabilityunder a suitable nonlinear change of variables.

Keywords : uniformly global asymptotic stability; exponential stabilityClassification :

∗34D23 Global stability34A25 Analytical theory of ODE

Zbl 0963.65138

Falcone, Maurizio; Grune, Lars; Wirth, FabianA maximum time approach to the computation of robust domains of attrac-tion. (English)Fiedler, B. (ed.) et al., International conference on differential equations. Proceedingsof the conference, Equadiff ’99, Berlin, Germany, August 1-7, 1999. Vol. 2. Singapore:World Scientific. 844-849 (2000). ISBN 981-02-4989-6/hbk

Summary: We present an optimal control based algorithm for the computation of robustdomains of attraction for perturbed systems. We give a sufficient condition for thecontinuity of the optimal value function and a characterization by Hamilton-Jacobiequations. A numerical scheme is presented and illustrated by an example.

Keywords : numerical example; dynamical systems; stability; optimal control; algo-rithm; domains of attraction; Hamilton-Jacobi equationsClassification :

∗65P40 Nonlinear stabilities37C70 Attractors and repellers, topological structure37C75 Stability theory

Zbl 0960.65137

Grune, LarsConvergence rates of perturbed attracting sets with vanishing perturbation.(English)J. Math. Anal. Appl. 244, No.2, 369-392 (2000). ISSN 0022-247Xhttp://dx.doi.org/10.1006/jmaa.2000.6707http://www.sciencedirect.com/science/journal/0022247X

The author starts up an analysis of the rate of convergence for attracting sets witharbitrary rate of attraction. Necessary and sufficient conditions for certain rates of con-vergence are derived. They can be seen as generalizations of the exponential properties.Some applications, concerning the estimates for the semicontinuous discretization errorfor one step discretization of arbitrary sets are carried out.

Calin Ioan Gheorghiu (Cluj-Napoca)Keywords : time varying perturbations; convergence; exponentially attracting sets

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Classification :∗65P20 Numerical chaos37C70 Attractors and repellers, topological structure37M15 Symplectic integrators

Zbl 0958.93077

Grune, LarsHomogeneous state feedback stabilization of homogenous systems. (English)SIAM J. Control Optimization 38, No.4, 1288-1308 (2000). ISSN 0363-0129; ISSN1095-7138http://dx.doi.org/10.1137/S0363012998349303http://epubs.siam.org/sam-bin/dbq/article/34930http://epubs.siam.org/sam-bin/dbq/toclist/SICON

The paper gives a solution to the problem of finding a homogeneous feedback for sta-bilizing homogeneous systems. For asymptotically null controllable systems, in general,standard feedback concepts do not allow this. Lyapunov exponents and a kind of dis-cretized feedback are used here to obtain a positive answer. The paper is the culminationof a series of papers by the author where the necessary tools are developed. The analyt-ical results are illustrated by two (numerical) examples, one of them being Brockett’sexample of the nonholonomic integrator for which no continuous stabilizing feedbackexists.

Fritz Colonius (Augsburg)Keywords : homogeneous feedback; control Lyapunov function; Lyapunov exponents;stabilization; discretized feedbackClassification :

∗93D15 Stabilization of systems by feedback93D30 Scalar and vector Lyapunov functions34D08 Lyapunov exponents

Zbl 1043.93549

Grune, Lars; Sontag, Eduardo D.; Wirth, Fabian R.Asymptotic stability equals exponential stability, and ISS equals finite en-ergy gain – if you twist your eyes. (English)Syst. Control Lett. 38, No. 2, 127-134 (1999). ISSN 0167-6911http://dx.doi.org/10.1016/S0167-6911(99)00053-5http://www.sciencedirect.com/science/journal/01676911

Summary: In this paper we show that uniformly global asymptotic stability for a familyof ordinary differential equations is equivalent to uniformly global exponential stabilityunder a suitable nonlinear change of variables. The same is shown for input-to-state

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stability and input-to-state exponential stability, and for input-to-state exponential sta-bility and a nonlinear H∞ estimate.

Classification :∗93D20 Asymptotic stability of control systems34D20 Lyapunov stability of ODE93D05 Lyapunov and other classical stabilities of control systems

Zbl 0970.65073

Grune, L.; Metscher, M.; Ohlberger, M.On numerical algorithm and interactive visualization for optimal controlproblems. (English)Comput. Vis. Sci. 1, No.4, 221-229 (1999). ISSN 1432-9360; ISSN 1433-0369http://dx.doi.org/10.1007/s007910050020http://www.springerlink.com/content/100525/

Summary: We present methods for the visualization of the numerical solution of opti-mal control problems. The solution is based on dynamic programming techniques wherethe corresponding optimal value function is approximated on an adaptively refined grid.This approximation is then used in order to compute approximately optimal solutiontrajectories. We discuss requirements for the efficient visualization of both the optimalvalue functions and the optimal trajectories and develop graphic routines that in partic-ular support adaptive, hierarchical grid structures, interactivity and animation. Severalimplementational aspects using the Graphics Programming Environment ‘GRAPE’ arediscussed.

Keywords : algorithm; visualization; optimal control; dynamic programming; optimalsolution trajectoriesClassification :

∗65K10 Optimization techniques (numerical methods)49L20 Dynamic programming method (infinite-dimensional problems)68U05 Computational geometry, etc.

Zbl 0948.93051

Grune, LarsInput-to-state stability of exponentially stabilized semilinear control systemswith inhomogeneous perturbations. (English)Syst. Control Lett. 38, No.1, 27-35 (1999). ISSN 0167-6911http://dx.doi.org/10.1016/S0167-6911(99)00044-4http://www.sciencedirect.com/science/journal/01676911http://www.sciencedirect.com/science/journal/01676911

Summary: We investigate the robustness of state feedback stabilized semilinear systemssubject to inhomogeneous perturbations in terms of input-to-state stability. We con-

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sider a general class of exponentially stabilizing feedback controls which covers sampleddiscrete feedbacks and discontinuous mappings as well as classical feedbacks and derivea necessary and sufficient condition for the corresponding closed-loop systems to beinput-to-state stable with exponential decay and linear dependence on the perturba-tion. This condition is easy to check and admits a precise estimate for the constantsinvolved in the input-to-state stability formulation. Applying this result to an optimalcontrol based discrete feedback yields an equivalence between (open-loop) asymptoticnull controllability and robust input-to-state (state feedback) stabilizability.

Keywords : input-to-state stability; stabilizing feedback control; robustnessClassification :

∗93D21 Adaptive and robust stabilization93B05 Controllability93C10 Nonlinear control systems

Zbl 0945.93604

Grune, LarsStabilization by sampled and discrete feedback with positive sampling rate.(English)Aeyels, Dirk (ed.) et al., Stability and stabilization of nonlinear systems. Proceedingsof the 1st workshop on Nonlinear control network, held in Gent, Belgium, March 15-16,1999. London: Springer. Lect. Notes Control Inf. Sci. 246, 165-182 (1999). ISBN1-85233-638-2http://link.springer.de/link/service/series/0642/bibs/9246/92460165.htm

Classification :∗93D15 Stabilization of systems by feedback93C57 Sampled-data control systems

Zbl 0917.93058

Grune, Lars; Wirth, FabianFeedback stabilization of discrete-time homogeneous semi-linear systems.(English)Syst. Control Lett. 37, No.1, 19-30 (1999). ISSN 0167-6911http://dx.doi.org/10.1016/S0167-6911(98)00110-8http://www.sciencedirect.com/science/journal/01676911http://www.sciencedirect.com/science/journal/01676911

Summary: For discrete-time semi-linear systems satisfying an accessibility conditionasymptotic null controllability is equivalent to exponential feedback stabilizability usinga piecewise constant feedback. A constructive procedure that yields such a feedback ispresented.

Keywords : analytic discrete-time systems; forward accessibility; feedback stabilization;

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numerical schemeClassification :

∗93D15 Stabilization of systems by feedback93B40 Computational methods in systems theory93C55 Discrete-time control systems93C10 Nonlinear control systems

Zbl 0925.93835

Grune, L.; Wirth, F.Stabilization of discrete-time bilinear systems. (English)ZAMM, Z. Angew. Math. Mech. 78, Suppl. 3, S925-S926 (1998). ISSN 0044-2267;ISSN 1521-4001http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=5007542

Classification :∗93D15 Stabilization of systems by feedback93C55 Discrete-time control systems93C10 Nonlinear control systems

Zbl 0918.49002

Grune, LarsOn the relation between discounted and average optimal value functions.(English)J. Differ. Equations 148, No.1, 65-99 (1998). ISSN 0022-0396http://dx.doi.org/10.1006/jdeq.1998.3451http://www.sciencedirect.com/science/journal/00220396

The author investigates the relation between discounted and average deterministic op-timal control problems for nonlinear control systems. In particular, he is interested inthe corresponding optimal value functions. Using the concepts of viability, chain con-trollability, and controllability, a global convergence result for vanishing discount rate isobtained. Basic ingredients for the analysis are an Abelian type theorem, controllabil-ity properties of the system, and the Morse decomposition of the corresponding controlflow.

P.Neittaanmaki (Jyvaskyla)Keywords : deterministic nonlinear optimal control; discounted optimal control; aver-age optimal control; Morse decomposition; viable sets; chain control sets; control sets;viabilityClassification :

∗49J15 Optimal control problems with ODE (existence)49K15 Optimal control problems with ODE (nec./ suff.)

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Zbl 0910.93063

Grune, LarsAsymptotic controllability and exponential stabilization of nonlinear controlsystems at singular points. (English)SIAM J. Control Optimization 36, No.5, 1485-1503 (1998). ISSN 0363-0129; ISSN1095-7138http://dx.doi.org/10.1137/S0363012997315919http://epubs.siam.org/sam-bin/dbq/article/31591http://epubs.siam.org/sam-bin/dbq/toclist/SICON

The author studies the stabilization of nonlinear control systems on Rd ×M given by

x(t) = f(x(t), y(t), u(t)), y(t) = g(y(t), u(t)),

where x ∈ Rd and y ∈ M , M is a Riemannian manifold and f, g are vector fields whichare C2 in x, Lipschitz in y, and continuous in u. The control function u(·) may bechosen from the set U := u : R → U | u(·) measurable, where U ⊂ Rm is compact.The author’s interest lies in the stabilization of the x-component at a singular pointx∗, i.e., a point where f(x∗, y, u) = 0 for all (y, u) ∈ M × U. Such singular situationsdo typically occur if the control enters in the parameters of an uncontrolled system ata fixed point, for instance, when the restoring force of a nonlinear oscillator is con-trolled. By using a discrete feedback law, results on the relation between asymptoticcontrollability and exponential stabilization are developed and an equivalence theoremis derived. The construction of the feedback is obtained by minimizing the Lyapunovexponent of the linearized system. For semilinear systems, asymptotic null controlla-bility and exponential stabilizability by a discrete feedback turn out to be equivalent.For general nonlinear systems the equivalence between uniform exponential controlla-bility and uniform exponential stabilizability is shown. An example illustrates thatuniform exponential controllability is in fact a necessary condition for the applicabilityof linearization techniques. The obtained results can be applied to the stabilizationproblem of an inverted pendulum for which the suspension point is moved up and downperiodically and the period of this motion can be controlled.

Alexandr Vasil’ev (Odessa)Keywords : stabilization of nonlinear system by feedback; nonlinear control systems;singular points; Lyapunov exponents; discounted optimal control problems; discretefeedback control; robustness of discrete feedback control; interrelation between con-trollability and stabilization; uniform exponential controllabiltiy; uniform exponentialstabilizability; inverted pendulumClassification :

∗93D15 Stabilization of systems by feedback93C10 Nonlinear control systems34D08 Lyapunov exponents

Zbl 0880.65045

Grune, Lars

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An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation.(English)Numer. Math. 75, No.3, 319-337 (1997). ISSN 0029-599X; ISSN 0945-3245http://dx.doi.org/10.1007/s002110050241http://link.springer.de/link/service/journals/00211/

An adaptive grid scheme for the solution of the discrete first order Hamilton-Jacobi-Bellmann equation

supu∈U

vh(x)− βvh(Φh(x, u))− hg(x, u) = 0

on Ω ⊂ Rn with 0 < β < 1 is developed, where Φh is the right hand side of a discretetime control system and g is the cost function. Error estimates are proved and anadapting iteration for the discretization of the state space is developed.

H.Benker (Merseburg)Keywords : finite difference method; error estimates; adaptive grid scheme; Hamilton-Jacobi-Bellmann equation; discrete time control systemClassification :

∗65K10 Optimization techniques (numerical methods)49L25 Viscosity solutions49J20 Optimal control problems with PDE (existence)49M25 Finite difference methods

Zbl 0876.93032

Grune, LarsNumerical stabilization of bilinear control systems. (English)SIAM J. Control Optimization 34, No.6, 2024-2050 (1996). ISSN 0363-0129; ISSN1095-7138http://dx.doi.org/10.1137/S0363012994272290http://epubs.siam.org/sam-bin/dbq/toclist/SICON

Under investigation are bilinear control systems of the form

(1) x′(t) = (A0 +∑

ui(t)Ai)x(t), x(0) = x0.

Here x ∈ Rn, the Ai are constant matrices and the control u(t) is measurable withvalues in a given compact convex set with nonempty interior.A numerical approximation of the Lyapunov exponent is calculated and used for stabi-lizing the system by suitable controls.A related optimal control problem is considered with cost

(2) J =∫

g(x, u)dt.

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This is compared with a “discounted” problem

(3) Jδ =∫

e−δtg(x, u)dt

and an “averaged” problem

(4) J0 = lim sup(1/T )∫ T

0

g(x, u)dt.

Some results concerning their optimal value function were established earlier by F.Colonius [Math. Oper. Res. 14, 309-316 (1989; Zbl 0677.49025)] and F. Wirth [Math.Oper. Res. 18, 1006-1019 (1993; Zbl 0824.49001)]. Two approximation theorems aregiven here, together with some estimates and related convergence theorems. All ofthis is then applied to the bilinear system (1), leading to a numerical solution, usingdiscretization, for the discounted optimal control problem as an approximation of (1).Results of such numerical computations are included.

E.O.Roxin (Kingston/Rhode Island)Keywords : stabilization; bilinear control systems; Lyapunov exponent; optimal control;discretizationClassification :

∗93B40 Computational methods in systems theory93D15 Stabilization of systems by feedback34D08 Lyapunov exponents49L25 Viscosity solutions

Zbl 0867.93071

Grune, LarsDiscrete feedback stabilization of semilinear control systems. (English)ESAIM, Control Optim. Calc. Var. 1, 207-224 (1996). ISSN 1292-8119; ISSN 1262-3377http://dx.doi.org/10.1051/cocv:1996106http://www.edpsciences.org/articles/cocv/abs/1998/01/cocvEng-Vol1.7.htmlhttp://www.edpsciences.org/journal/index.cfm?edpsname=cocv

This paper deals with exponential stabilizability for semilinear control systems of theform x = A(u)x where the values of the control parameter u are constrained to a com-pact set U . The strategy proposed by the authors relies on the solution of a discountedoptimal control problem. This allows to define a feedback function F and a time steph. In turn, F and h define a state depending piecewise constant control law, whichprovides the desired stabilization policy. Finally, the author proposes some numericalapplications.

A.Bacciotti (Torino)Keywords : stabilization; exponential stabilizability; discounted optimal control; piece-wise constant controlClassification :

∗93D15 Stabilization of systems by feedback

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