Chapter8Open ProblemsIn this chapter,we would like to give a list of open and unanswered problems in Mathematical Control Theory.The solutions of these open problems will be very important for the development of modern nonlinear control theory.Expectedly novel mathematical analysis and synthesis tools need to be developed to address these challenging problems.The interested reader should also consult the book[3]for other significant and important open problems in Mathematical Control Theory.Open Problem#1Under what conditions WIOS implies IOS?A qualitative characterization of the IOS property for abstract control systems as discussed in this book has not been available yet.For systems described by ODEs, many qualitative characterizations of the ISS and IOS properties are provided in [21–23].Moreover,Theorem4.1in Chap.4gives a complete qualitative character-ization of the WIOS property:“0-GAOS”+“RFC”+“the continuity with respect to initial conditions and external inputs”implies WIOSA similar qualitative characterization for the IOS property in a general context of abstract dynamical systems as discussed in this book will be very important for control designs and applications.Open Problem#2Development of small-gain techniques for dynamical systems described by Partial Differential Equations(PDEs).Small-gain results have been well studied forfinite-dimensional nonlinear sys-tems described by ordinary differential,or difference,equations(see,e.g.,[8–10] and references therein).However,as of today,there is little research devoted to the development of small-gain techniques for nonlinear systems described by Partial Differential Equations(PDEs).We believe that the small-gain results provided in the present book(Theorems5.1and5.2in Chap.5)will pave the road for the appli-cation of small-gain results to systems described by PDEs.I.Karafyllis,Z.-P.Jiang,Stability and Stabilization of Nonlinear Systems,381 Communications and Control Engineering,DOI10.1007/978-0-85729-513-2_8,©Springer-Verlag London Limited2011Open Problem#3Formulas for the Coron–Rosier methodology.Theorem6.1in Chap.6is an existence-type result.Although its proof is con-structive,it cannot be easily applied for feedback design purposes.The creation of formulas for the Coron–Rosier approach will be very significant for control pur-poses,since the Coron–Rosier approach can allow nonconvex control sets and does not require additional properties for the Control Lyapunov Function.The signifi-cance of the solution of this open problem is also noted in[5].Open Problem#4When is a nonlinear,time-varying,time-delay system stabiliz-able?We have recently provided a positive answer to the above question when the sys-tem only involves state-delay[13].A complete answer to the question of when the nonlinear time-varying system with both state and input delays is stabilizable re-mains open and requires deeper investigation.Nonetheless,it should be mentioned that sufficient,but not necessary,conditions for the solution of the stabilization prob-lem with input delays are proposed in the recent work of Krsti´c[14–16](also see [11]).To our knowledge,a necessary and sufficient condition for stabilizability is missing even for linear time-varying systems with input delays.Open Problem#5Application of small-gain results for distributed feedback design of large-scale nonlinear systems.Large-scale systems are abundant in variousfields of science and engineering and have gained increasing attention due to emerging engineering and biomedical applications.Examples of these applications are from smart grids with green and re-newable energy sources,modern transportation networks,and biological networks. There has been some success with the use of decentralized control strategy for both linear and nonlinear large-scale systems;see[7,19]and many references therein. Clearly more remains to be accomplished in this excitingfield.We feel that small-gain is a very appropriate tool for addressing some of these modern-day challenges. The small-gain results of the present book(Theorems5.1and5.2in Chap.5)make a preliminary step forward toward studying some complex large-scale systems be-yond the past literature of decentralized systems and control.Open Problem#6Extension of the discretization approach for autonomous sys-tems.The discretization approach for Lyapunov functionals was described in Chap.2 (Propositions2.4and2.5).However,as remarked in Chap.2,the discretization ap-proach requires good knowledge of some approximation of the solution map,and its use has been restricted to time-varying systems with special structure(see[1,17, 18]).An extension of the discretization approach for autonomous systems wouldbe an important contribution in stability theory because such a result would al-low the use of positive definite functions with non sign-definite derivative.The re-quired extension of the discretization approach must utilize appropriate differential inequalities in the same spirit as the classical Lyapunov’s approach(without requir-ing knowledge of the solution map or a system with special structure).The recent work in[12]is an attempt in this research direction(see also references therein). However,the problem is still completely“untouched.”Open Problem#7Application of feedback design methodologies to other mathe-matical problems.In this book,we have seen the applications of certain tools of modern nonlinear control theory to problems arising from mathematics and economics.Particularly, we have seen•applications of small-gain results to game theory(see Sect.5.5in Chap.5),•applications to numerical analysis(see Sect.7.3).We believe that feedback design methodologies can be applied with success to other areas of mathematical sciences.Fixed Point Theory(see[6])and Optimization Theory can be benefited by the application of certain tools of modern nonlinear con-trol theory.Corollary5.4in Chap.5already shows that small-gain results can have serious consequences in Fixed Point Theory.Further connections between Fixed Point Theory and Stability Theory are provided by the work of Burton(see[4]and references therein)but are in the opposite direction from what we propose,that is, the work of Burton applies results from Fixed Point Theory to Stability Theory.The efforts for the solution of problems in Game Theory,Numerical Analysis, Fixed Point Theory,and Optimization Theory will necessarily demand the creation of novel results in stability theory and feedback stabilization theory.Therefore,the application of modern nonlinear control theory to other areas of applied mathe-matics will result to a“knowledge feedback mechanism”between Mathematical Control Theory and other areas in mathematics!Open Problem#8Integral input-to-state stability(for short,iISS)in complex dy-namical systems.The external stability results of this book are exclusively targeted at extensions of Sontag’s ISS property and its variants to a very general context of complex dynamic systems.That is,we want to address a wide class of dynamical systems which may not satisfy the semigroup property,motivated by important examples of hybrid sys-tems,switched systems,and time-delay systems.It remains an open and important, but interesting,question to know how much we could do with the iISS property introduced in[2,20].References1.Aeyels,D.,Peuteman,J.:A new asymptotic stability criterion for nonlinear time-variant dif-ferential equations.IEEE Transactions on Automatic Control43(7),968–971(1998)2.Angeli,D.,Sontag,E.D.,Wang,Y.:A characterization of integral input-to-state stability.IEEETransactions on Automatic Control45(6),1082–1097(2000)3.Blondel,V.D.,Megretski,A.(eds.):Unsolved Problems in Mathematical Systems and ControlTheory.Princeton University Press,Princeton(2004)4.Burton,T.A.:Stability by Fixed Point Theory for Functional Differential Equations.Dover,Mineola(2006)5.Coron,J.-M.:Control and Nonlinearity.Mathematical Surveys and Monographs,vol.136.AMS,Providence(2007)6.Granas,A.,Dugundji,J.:Fixed Point Theory.Springer Monographs in Mathematics.Springer,New York(2003)7.Jiang,Z.P.:Decentralized control for large-scale nonlinear systems:A review of recent results.Dynamics of Continuous,Discrete and Impulsive Systems11,537–552(2004).Special Issue in honor of Prof.Siljak’s70th birthday8.Jiang,Z.P.:Control of interconnected nonlinear systems:a small-gain viewpoint.In:deQueiroz,M.,Malisoff,M.,Wolenski,P.(eds.)Optimal Control,Stabilization,and Nonsmooth Analysis.Lecture Notes in Control and Information Sciences,vol.301,pp.183–195.Springer, Heidelberg(2004)9.Jiang,Z.P.,Mareels,I.M.Y.:A small-gain control method for nonlinear cascaded systems withdynamic uncertainties.IEEE Transactions on Automatic Control42,292–308(1997)10.Jiang,Z.P.,Teel,A.,Praly,L.:Small-gain theorems for ISS systems and applications.Mathe-matics of Control,Signals,and Systems7,95–120(1994)11.Karafyllis,I.:Stabilization by means of approximate predictors for systems with delayed in-put.To appear in SIAM Journal on Control and Optimization12.Karafyllis,I.:Can we prove stability by using a positive definite function with non sign-definite derivative?Submitted to Nonlinear Analysis Theory,Methods and Applications 13.Karafyllis,I.,Jiang,Z.P.:Necessary and sufficient Lyapunov-like conditions for robustnonlinear stabilization.ESAIM:Control,Optimization and Calculus of Variations(2009).doi:10.1051/cocv/2009029,pp.1–42,August200914.Krsti´c,M.:Delay Compensation for Nonlinear,Adaptive,and PDE Systems.Systems&Con-trol:Foundations&Applications.Birkhäuser,Boston(2009)15.Krsti´c,M.:Input delay compensation for forward complete and feedforward nonlinear sys-tems.IEEE Transactions on Automatic Control55,287–303(2010)16.Krsti´c,M.:Lyapunov stability of linear predictor feedback for time-varying input delay.IEEETransactions on Automatic Control55,554–559(2010)17.Peuteman,J.,Aeyels,D.:Exponential stability of slowly time-varying nonlinear systems.Mathematics of Control,Signals and Systems15,42–70(2002)18.Peuteman,J.,Aeyels,D.:Exponential stability of nonlinear time-varying differential equationsand partial averaging.Mathematics of Control,Signals and Systems15,202–228(2002)19.Siljak,D.:Decentralized Control of Complex Systems.Academic Press,New York(1991)20.Sontag,E.D.:Comments on integral variants of ISS.Systems Control Letters3(1–2),93–100(1998)21.Sontag,E.D.,Wang,Y.:On characterizations of the input-to-state stability property.Systemsand Control Letters24,351–359(1995)22.Sontag,E.D.,Wang,Y.:New characterizations of the input-to-state stability.IEEE Transac-tions on Automatic Control41,1283–1294(1996)23.Sontag,E.D.,Wang,Y.:Lyapunov characterizations of input to output stability.SIAM Journalon Control and Optimization39,226–249(2001)。
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